Analysis/synthesis windowing function for modulated lapped transformation

ABSTRACT

There are provided methods and apparatus for performing modified cosine transformation (MDCT) with an analysis/synthesis windowing function, using an analysis windowing function having a meandering portion which passes a linear function in correspondence of at least four points.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is a continuation of copending InternationalApplication No. PCT/EP2018/080532, filed Nov. 8, 2018, which isincorporated herein by reference in its entirety, and additionallyclaims priority from European Application No. EP 17201086.0, filed Nov.10, 2017, which is incorporated herein by reference in its entirety.

BACKGROUND OF THE INVENTION

Lapped transforms have been developed for several audiocodingapplications. These transforms are normally performed on consecutiveblocks of a larger dataset (e.g., two consecutive frames) for an audiosignal. Subsequent blocks may be overlapped. Accordingly, a last part ofone block coincides with a first part of a subsequent block. A windowingfunction may be used.

Asymmetric modified discrete cosine transform (MDCT) windows, asymmetricmodified discrete sine transform (MDST), time to frequencytransformations, have been developed which embody modulated lappedtransforms.

Asymmetric modified discrete cosine transform (MDCT) windows, asymmetricmodified discrete sine transform (MDST) windows and other types ofwindows have been developed in the past years as they provide improvedfrequency responses compared to symmetric shapes, esp. for low delaysignal applications, such as audio coding applications.

The first generation of window shapes just focused on the design of thefrequency response, e.g. the G.718 or MPEG-4 AAC-ELD window. Recentdevelopments also take the temporal shape of the window into accountwhich is responsible for the temporal modulation of the quantizationerror, e.g. the ALDO window as used in the 3GPP EVS codec.

However, the design approach presented there comes along with theproblem that the window shapes show a non-continuous differentiationwhich leads to a suboptimal frequency response. The described inventionin this document presents a solution to overcome this problem.

SUMMARY

According to an embodiment, an apparatus for encoding an informationsignal including a plurality of frames may have:

-   -   a modulated lapped transform tool for transforming a time        domain, TD, representation of the information signal, or a        processed version thereof, into a frequency domain, FD,        representation, the modulated lapped transform tool being        configured to perform a modulated lapped transform analysis        using an analysis windowing function having a meandering portion        crossing a linear function in correspondence of at least four        points; and    -   a bitstream writer configured to prepare a bitstream based on        the FD representation of the information signal or a processed        version thereof,    -   wherein the analysis windowing function is defined so as to be        asymmetric,    -   wherein the analysis windowing function is defined so as to be,        in the meandering portion:        -   greater than the linear function in a first interval between            a first crossing point and a second crossing point;        -   lower than the linear function in a second interval between            the second crossing point and a third crossing point;        -   greater than the linear function in a third interval between            the third crossing point and a fourth crossing point,    -   wherein the analysis windowing function is defined so that the        absolute maximum value is in one of the first and third        interval,    -   wherein the linear function is a constant function with constant        value 1.

According to another embodiment, a method for encoding an informationsignal including a plurality of frames may have the steps of:

-   -   performing a modified cosine transformation, MDCT, analysis for        transforming a time domain, TD, representation of the        information signal, or a processed version thereof, into a        frequency domain, FD, representation, using an analysis        windowing function having a meandering portion which passes a        linear function in correspondence of at least four points,    -   wherein the analysis windowing function is defined so as to be        asymmetric,    -   wherein the analysis windowing function is defined so as to be,        in the meandering portion:        -   greater than the linear function in a first interval between            a first crossing point and a second crossing point;        -   lower than the linear function in a second interval between            the second crossing point and a third crossing point;        -   greater than the linear function in a third interval between            the third crossing point and a fourth crossing point,    -   wherein the analysis windowing function is defined so that the        absolute maximum value is in one of the first and third        interval,    -   wherein the linear function is a constant function with constant        value 1.

According to another embodiment, a non-transitory digital storage mediummay have a computer program stored thereon to perform the inventivemethod, when said computer program is run by a computer.

In accordance to an aspect there is provided an apparatus for encodingan information signal comprising a plurality of frames, the apparatuscomprising:

-   -   a modulated lapped transform tool for transforming a time        domain, TD, representation of the information signal, or a        processed version thereof, into a frequency domain, FD,        representation, the modulated lapped transform tool being        configured to perform a modulated lapped transform analysis        using an analysis windowing function having a meandering portion        crossing a linear function in correspondence of at least four        points and    -   a bitstream writer configured to prepare a bitstream based on        the FD representation of the information signal or a processed        version thereof,    -   wherein the analysis windowing function is defined so as to be        asymmetric,    -   wherein the analysis windowing function is defined so as to be,        in the meandering portion        -   greater than the linear function in a first interval between            a first crossing point and a second crossing point;        -   lower than the linear function in a second interval between            the second crossing point and a third crossing point;        -   greater than the linear function in a third interval between            the third crossing point and a fourth crossing point,    -   wherein the analysis windowing function is defined so that the        absolute maximum value is in one of the first and third        interval,    -   wherein the linear function is a constant function with constant        value 1.

The analysis windowing function may be defined so that the maximum ofthe analysis windowing function is less than 25% greater than the valueof the linear function at the same time instant.

The apparatus for encoding an information signal may comprise aplurality of frames, the apparatus may comprise:

-   -   a modulated lapped transform tool for transforming a time        domain, TD, representation of the information signal, or a        processed version thereof, into a frequency domain, FD,        representation, the modulated lapped transform tool being        configured to perform a modulated lapped transform analysis        using an analysis windowing function having a meandering portion        crossing a linear function in correspondence of at least four        points; and    -   a bitstream writer configured to prepare a bitstream based on        the FD representation of the information signal or a processed        version thereof,    -   wherein the analysis windowing function is defined so as to be        asymmetric,    -   wherein the analysis windowing function is defined so that the        maximum of the analysis windowing function is less than 25%        greater than the value of the linear function at the same time        instant,    -   wherein the linear function is a constant function.

The apparatus for encoding an information signal may comprise aplurality of frames, the apparatus may comprise:

-   -   a modulated lapped transform tool for transforming a time        domain, TD, representation of the information signal, or a        processed version thereof, into a frequency domain, FD,        representation, the modulated lapped transform tool being        configured to perform a modulated lapped transform analysis        using an analysis windowing function having a meandering portion        crossing a linear function in correspondence of at least four        points; and    -   a bitstream writer configured to prepare a bitstream based on        the FD representation of the information signal or a processed        version thereof.

The apparatus may comprise the modulated lapped transform tool isconfigured to:

-   -   scale time input buffers and/or cosine or sine values with        values of the analysis windowing function.

The apparatus may comprise the modulated lapped transform tool isconfigured to use input buffers in the form of

t(n)=x(Z−N _(F) +n) for n=0 . . . 2N _(F)−1−Z, and

t(2N−Z+n)=0 for n=0 . . . Z−1

wherein x(n) is a TD sample of the information signal or a processedversion of the information signal at the time instant n, N_(F) is thenumber of samples processed in one frame, and Z is the number of leadingzeros in modulated lapped transform window.

The apparatus may comprise the modulated lapped transform tool that isconfigured to perform:

${X(k)} = {\sqrt{\frac{2}{N_{F}}}{\sum\limits_{n = 0}^{{2N_{F}} - 1}{{{w_{N}(n)} \cdot {t(n)}}{\cos \left\lbrack {\frac{\pi}{N_{F}}\left( {n + \frac{1}{2} + \frac{N_{F}}{2}} \right)\left( {k + \frac{1}{2}} \right)} \right\rbrack}}}}$for  k = 0  …  N_(F) − 1

where X(k) is the modulated lapped transform frequency value at afrequency index k, n is the time instant, w_(N) (n) is the analysiswindowing function, t (n) is a time input buffer, and N_(F) is thenumber of samples processed in one frame.

The apparatus for decoding an information signal, or a processed versionthereof, defined in the frequency domain, FD, the apparatus maycomprise:

-   -   a bitstream reader configured to read a bitstream encoding the        information signal; and    -   an inverse modulated lapped transform tool configured to perform        an inverse modulated lapped transform synthesis on the        information signal, or a processed version thereof, using a        synthesis windowing function having a meandering portion        crossing a linear function in correspondence of at least four        points,    -   wherein the synthesis windowing function is defined so as to be        asymmetric,    -   wherein the synthesis windowing function is defined so as to be,        in the meandering portion:        -   greater than the linear function in a first interval between            a first crossing point and a second crossing point;        -   lower than the linear function in a second interval between            the second crossing point and a third crossing point;        -   greater than the linear function in a third interval between            the third crossing point and a fourth crossing point,    -   wherein the synthesis windowing function is defined so that the        absolute maximum value is in one of the first and third        interval,    -   wherein the linear function is a constant function with constant        value 1.

The apparatus may comprise

-   -   the synthesis windowing function that defined so that the        maximum of the synthesis windowing function is less than 25%        greater than the value of the linear function at the same time        instant.

The apparatus for decoding an information signal, or a processed versionthereof, defined in the frequency domain, FD, the apparatus maycomprise:

-   -   a bitstream reader configured to read a bitstream encoding the        information signal; and    -   an inverse modulated lapped transform tool configured to perform        an inverse modulated lapped transform synthesis of the        information signal, or a processed version thereof, using a        synthesis windowing function having a meandering portion        crossing a linear function in correspondence of at least four        points,    -   wherein the synthesis windowing function is defined so as to be        asymmetric,    -   wherein the synthesis windowing function is defined so that the        maximum of the synthesis windowing function is less than 25%        greater than the value of the linear function at the same time        instant,    -   wherein the linear function is a constant function.

An apparatus for decoding an information signal, or a processed versionthereof, defined in the frequency domain, FD, the apparatus maycomprise:

-   -   a bitstream reader configured to read a bitstream encoding the        information signal; and    -   an inverse modulated lapped transform tool configured to perform        an inverse modulated lapped transform synthesis on the        information signal, or a processed version thereof, using a        synthesis windowing function having a meandering portion        crossing a linear function in correspondence of at least four        points.

The apparatus wherein the inverse modulated lapped transform tool may beconfigured to:

-   -   scale values at a time domain aliasing buffer with values of the        synthesis windowing function.

The apparatus wherein the inverse modulated lapped transform tool may beconfigured to generate a time domain, TD, signal representation in theform of

${\hat{t}(n)} = {\sqrt{\frac{2}{N_{F}}}{\sum\limits_{k = 0}^{N_{F} - 1}{{\hat{X}(k)}{\cos \left\lbrack {\frac{\pi}{N_{F}}\left( {n + \frac{1}{2} + \frac{N_{F}}{2}} \right)\left( {k + \frac{1}{2}} \right)} \right\rbrack}}}}$for  n = 0  …  2N_(F) − 1

wherein {circumflex over (t)}(n) is an aliasing buffer, {circumflex over(X)}(k) is the information signal or a processed version thereof, andN_(F) is the number of samples for a TD frame.

The apparatus wherein the inverse modulated lapped transform tool may beconfigured to:

-   -   perform a windowing of the time-aliased buffer.

The apparatus wherein the inverse modulated lapped transform tool may beconfigured to perform a windowing operation by performing:

{circumflex over (t)}(n)=w _(N)(2N−1−n)·{circumflex over (t)}(n) for n=0. . . 2N _(F)−1

The apparatus wherein the inverse modulated lapped transform tool may beconfigured to perform an overlap-and-add operation.

The apparatus wherein the inverse modulated lapped transform tool may beconfigured to perform an overlap-and-add operation as:

{circumflex over (x)}(n)=mem_ola_add(n)+{circumflex over (t)}(Z+n) forn=0 . . . N _(F) −Z−1

{circumflex over (x)}(n)={circumflex over (t)}(Z+n) for n=N _(F) −Z . .. N _(F)−1

mem_ola_add(n)={circumflex over (t)}(N _(F) +Z+n) for n=0 . . . N _(F)−Z−1

wherein {circumflex over (x)}(n) is the output value, {circumflex over(t)}(.) is a windowed time-aliasing buffer, and N_(F) is the number ofsamples in one frame.

The apparatus may be so that:

-   -   the analysis windowing function and/or the synthesis windowing        function may be defined so as to be, in the meandering portion:        -   greater than the linear function in a first interval between            a first crossing point and a second crossing point;        -   lower than the linear function in a second interval between            the second crossing point and a third crossing point;        -   greater than the linear function in a third interval between            the third crossing point and a fourth crossing point.

The apparatus may be so that

-   -   the analysis windowing function and/or the synthesis windowing        function may be defined so that the absolute maximum value is in        the first or third interval.

The apparatus may be so that

-   -   the analysis windowing function and/or the synthesis windowing        function may be defined so that, in the meandering portion, a        relative maximum value is in the first or third interval and a        relative minimum value is in the second interval.

The apparatus may be so that

-   -   the analysis windowing function and/or the synthesis windowing        function may be defined so that, in the meandering portion, the        value of the meandering window function in correspondence to at        least one of the first and third interval is greater than 0.9.

The apparatus may be so that

-   -   the analysis windowing function and/or the synthesis windowing        function may be defined so as to present, in the meandering        portion, a value greater than the linear function in an interval        comprised of the 30% and 50% of two frames.

The apparatus may be so that

-   -   the analysis windowing function and/or the synthesis windowing        function may be defined so that the maximum of the analysis        windowing function and/or the synthesis windowing function is        less than 25% greater than the value of the linear function at        the same time instant.

The apparatus may be so that

-   -   the analysis windowing function and/or the synthesis windowing        function may be defined so that the maximum of the analysis        windowing function and/or the synthesis windowing function is        less than 5% greater than the value of the linear function at        the same time instant.

The apparatus may be so that

-   -   the analysis windowing function and/or the synthesis windowing        function may be defined so as to present a second numerical        differentiation between −3*10⁻⁴ and +3*10⁻⁴.

The apparatus may be so that

-   -   the analysis windowing function and/or the synthesis windowing        function may be defined so as to present a third numerical        differentiation between −2*10⁻⁵ and +2*10⁻⁵.

The apparatus may be so that

-   -   the analysis windowing function and/or synthesis windowing        function is defined so as to present a first numerical        differentiation between −0.01 and +0.01.

The apparatus may be so that

the analysis windowing function and/or synthesis windowing function isdefined so as to present a second numerical differentiation between−10⁻⁴ and +10⁻⁴.

The apparatus may be so that

the analysis windowing function and/or synthesis windowing function isdefined so as to present a second numerical differentiation between−10⁻⁵ and +10⁻⁵.

The apparatus may be so that

-   -   the linear function may be defined so as to be a non-increasing        function.

The apparatus may be so that

-   -   the linear function may be defined so as to be a strictly        decreasing function.

The apparatus may be so that

-   -   the linear function may be defined so as to be a non-decreasing        function.

The apparatus may be so that

-   -   the linear function may be defined so as to be a strictly        increasing function.

The apparatus may be so that

-   -   the linear function may be defined so as to have a value which        is constant or varies of maximum +2% or −2%.

The apparatus may be so that

-   -   the linear function may be defined so as to have increments        between −0.05 and −0.20.

The apparatus may be so that

-   -   the analysis windowing function and/or the synthesis windowing        function may be defined so as to be asymmetric.

The apparatus may be so that

-   -   the analysis windowing function and the synthesis windowing        function may be defined to be time reversed versions of each        other.

The apparatus may comprise:

-   -   a storage space to store the values of the analysis windowing        function and/or the synthesis windowing function.

The apparatus wherein the modified lapped transform may be a modifieddiscrete cosine transform, MDCT, or a modified discrete sine transform,MDST, and the inverse modified lapped transform is an inverse modifieddiscrete cosine transform, IMDCT, or inverse modified discrete sinetransform, IMDST.

The apparatus may be so that

-   -   the analysis windowing function and/or the synthesis windowing        function may be a constant function with constant value 1.

A system which may comprise:

-   -   an encoder as    -   a decoder as the apparatus.

The system wherein the encoder may comprise a communication unitconfigured to transmit a bitstream and/or the decoder comprises acommunication unit configured to receive a bitstream.

According to an aspect there is provided a method which comprisesperforming a modified cosine transformation, MDCT, analysis fortransforming a time domain, TD, representation of an information signal,or a processed version thereof, into a frequency domain, FD,representation, using an analysis windowing function having a meanderingportion which passes a linear function in correspondence of at leastfour points.

According to an aspect there is provided a method which comprisesperforming a modified cosine transformation, MDCT, analysis fortransforming a time domain, TD, representation of an information signal,or a processed version thereof, into a frequency domain, FD,representation, using an analysis windowing function having a meanderingportion which passes a linear function in correspondence of at leastfour points,

-   -   wherein the analysis windowing function is defined so as to be        asymmetric,    -   wherein the analysis windowing function is defined so as to be,        in the meandering portion:        -   greater than the linear function in a first interval between            a first crossing point and a second crossing point;        -   lower than the linear function in a second interval between            the second crossing point and a third crossing point;        -   greater than the linear function in a third interval between            the third crossing point and a fourth crossing point,    -   wherein the analysis windowing function is defined so that the        absolute maximum value is in one of the first and third        interval,    -   wherein the linear function is a constant function with constant        value 1.

According to an aspect there is provided a method which comprisesperforming a modified cosine transformation, MDCT, analysis fortransforming a time domain, TD, representation of an information signal,or a processed version thereof, into a frequency domain, FD,representation, using an analysis windowing function having a meanderingportion which passes a linear function in correspondence of at leastfour points,

-   -   wherein the analysis windowing function is defined so as to be        asymmetric,    -   wherein the analysis windowing function is defined so that the        maximum of the analysis windowing function is less than 25%        greater than the value of the linear function at the same time        instant,    -   wherein the linear function is a constant function.

According to an aspect there is provided a method which comprisesperforming a modified cosine transformation, MDCT, synthesis fortransforming a frequency domain, FD, representation of an informationsignal, or a processed version thereof, into a time domain, TD,representation, using a synthesis windowing function having a meanderingportion which passes a linear function in correspondence of at leastfour points.

According to an aspect there is provided a method which comprisesperforming a modified cosine transformation, MDCT, synthesis fortransforming a frequency domain, FD, representation of an informationsignal, or a processed version thereof, into a time domain, TD,representation, using a synthesis windowing function having a meanderingportion which passes a linear function in correspondence of at leastfour points,

-   -   wherein the synthesis windowing function is defined so as to be,        in the meandering portion:        -   greater than the linear function in a first interval between            a first crossing point and a second crossing point;        -   lower than the linear function in a second interval between            the second crossing point and a third crossing point;        -   greater than the linear function in a third interval between            the third crossing point and a fourth crossing point,    -   wherein the synthesis windowing function is defined so that the        absolute maximum value is in one of the first and third        interval,    -   wherein the linear function is a constant function with constant        value 1.

According to an aspect there is provided a method comprising performinga modified cosine transformation, MDCT, synthesis for transforming afrequency domain, FD, representation of an information signal, or aprocessed version thereof, into a time domain, TD, representation, usinga synthesis windowing function having a meandering portion which passesa linear function in correspondence of at least four points,

-   -   wherein the synthesis windowing function is defined so as to be        asymmetric,    -   wherein the synthesis windowing function is defined so that the        maximum of the synthesis windowing function is less than 25%        greater than the value of the linear function at the same time        instant,    -   wherein the linear function is a constant function.

A non-transitory storage unit storing instructions which, when runningon a processor, may cause the processor to execute a method.

There is provided an apparatus for encoding an information signalcomprising a plurality of frames, the apparatus comprising:

-   -   a modified discrete cosine transformation, MDCT, tool for        transforming a time domain, TD, representation of the        information signal, or a processed version thereof, into a        frequency domain, FD, representation, the MDCT tool being        configured to perform an MDCT analysis using an analysis        windowing function having a meandering portion crossing a linear        function in correspondence of at least four points; and    -   a bitstream writer configured to prepare a bitstream based on        the FD representation of the information signal or a processed        version thereof.

In examples, the MDCT tool is configured to scale time input buffersand/or cosine values with values of the analysis windowing function.

In examples, the MDCT tool is configured to use input buffers in theform of

t(n)=x(Z−N _(F) +n) for n=0 . . . 2N _(F)−1−Z,

wherein x(n) is a TD sample of the information signal or a processedversion of the information signal at the time instant n, N_(F) is thenumber of samples processed in one frame, and Z is the number of leadingzeros in MDCT window.

In examples, an initialization may be performed as:

t(2N−Z+n)=0 for n=0 . . . Z−1

In examples, the MDCT (or MDST) tool is configured to perform:

${X(k)} = {\sqrt{\frac{2}{N_{F}}}{\sum\limits_{n = 0}^{{2N_{F}} - 1}{{{w_{N}(n)} \cdot {t(n)}}{\cos \left\lbrack {\frac{\pi}{N_{F}}\left( {n + \frac{1}{2} + \frac{N_{F}}{2}} \right)\left( {k + \frac{1}{2}} \right)} \right\rbrack}}}}$for  k = 0  …  N_(F) − 1

where X(k) is the MDCT (or MDST) frequency value at a frequency index k,n is the time instant, W_(N)(n) is the analysis windowing function, t(n)is a time input buffer, and N_(F) is the number of samples processed inone frame.

In examples, there is provided an apparatus which comprises a bitstreamreader configured to read a bitstream encoding the information signal;and

-   -   an inverse modified cosine transformation, IMDCT (or IMDST),        tool, configured to perform an IMDCT (or IMDST) synthesis on the        information signal, or a processed version thereof, using a        synthesis windowing function having a meandering portion        crossing a linear function in correspondence of at least four        points.

In examples, the IMDCT (or IMDST) tool is configured to scale values ata time domain aliasing buffer with values of the synthesis windowingfunction.

In examples, the IMDCT (or IMDST) tool is configured to generate a timedomain, TD, signal representation in the form of

${\hat{t}(n)} = {\sqrt{\frac{2}{N_{F}}}{\sum\limits_{k = 0}^{N_{F} - 1}{{\hat{X}(k)}{\cos \left\lbrack {\frac{\pi}{N_{F}}\left( {n + \frac{1}{2} + \frac{N_{F}}{2}} \right)\left( {k + \frac{1}{2}} \right)} \right\rbrack}}}}$for  n = 0  …  2N_(F) − 1

wherein {circumflex over (t)}(n) is an aliasing buffer, {circumflex over(X)}(k) is the information signal or a processed version thereof, andN_(F) is the number of samples for a TD frame.

In examples, the IMDCT (or IMDST) tool is configured to perform awindowing of the time-aliased buffer.

In examples, the IMDCT (or IMDST) tool is configured to perform awindowing operation by performing:

{circumflex over (t)}(n)=w _(N)(2N−1−n)·{circumflex over (t)}(n) for n=0. . . 2N _(F)−1

In examples, the IMDCT (or IMDST) tool is configured to perform anoverlap-and-add operation, which may be, for example, as:

{circumflex over (x)}(n)=mem_ola_add(n)+{circumflex over (t)}(Z+n) forn=0 . . . N _(F) −Z−1

{circumflex over (x)}(n)={circumflex over (t)}(Z+n) for n=N _(F) −Z . .. N _(F)−1

mem_ola_add(n)={circumflex over (t)}(N _(F) +Z+n) for n=0 . . . N _(F)−Z−1

wherein {circumflex over (x)}(n) is the output value, {circumflex over(t)}(.) is a windowed time-aliasing buffer, and N_(F) is the number ofsamples in one frame.

In examples, the analysis windowing function and/or the synthesiswindowing function is defined so as to be, in the meandering portion:

-   -   greater than the linear function in a first interval between a        first crossing point and a second crossing point;    -   lower than the linear function in a second interval between the        second crossing point and a third crossing point;    -   greater than the linear function in a third interval between the        third crossing point and a fourth crossing point.

In examples, the analysis windowing function and/or the synthesiswindowing function is defined so that the absolute maximum value is inthe first or third interval.

In examples, the analysis windowing function and/or the synthesiswindowing function is defined so that, in the meandering portion, arelative maximum value is in the first or third interval and a relativeminimum value is in the second interval.

In examples, the analysis windowing function and/or the synthesiswindowing function is defined so that, in the meandering portion, thevalue of the meandering window function in correspondence to at leastone of the first and third interval is greater than 0.9.

In examples, the analysis windowing function and/or the synthesiswindowing function is defined so as to present, in the meanderingportion, a value greater than the linear function in an intervalcomprised of the 30% and 50% of two frames.

In examples, the analysis windowing function and/or the synthesiswindowing function is defined so that the maximum of the analysiswindowing function and/or the synthesis windowing function is less than25% (in particular 5%) greater than the value of the linear function atthe same time instant.

In examples, the analysis windowing function and/or the synthesiswindowing function is defined so as to present a second numericaldifferentiation is between −3*10⁻⁴ and +3*10⁻⁴ and/or the thirdnumerical differentiation is between −2*10⁻⁵ and +2*10⁻⁵.

In examples, the linear function is defined so as to have a value whichis constant or varies of maximum +2% or −2%.

In examples, the linear function is defined so as to have incrementsbetween −0.05 and −0.20.

In examples, the analysis windowing function and/or the synthesiswindowing function is defined so as to be asymmetric.

In examples, the analysis windowing function and the synthesis windowingfunction is defined are symmetric with each other.

In examples, a storage space to store the values of the analysiswindowing function and/or the synthesis windowing function.

In examples, there is provided a system comprising:

-   -   an encoder as an apparatus as above and/or below; and    -   a decoder as an apparatus as above and/or below.

In examples, the encoder comprise a communication unit configured totransmit a bitstream and/or the decoder comprises a communication unitconfigured to receive a bitstream.

In examples, there is also provided a method comprising performing anMDCT (MDST), analysis for transforming a time domain, TD, representationof an information signal, or a processed version thereof, into afrequency domain, FD, representation, using an analysis windowingfunction having a meandering portion which passes a linear function incorrespondence of at least four points.

In examples, there is also provided a method comprising performing anMDCT (or MDST), synthesis for transforming a frequency domain, FD,representation of an information signal, or a processed version thereof,into a time domain, TD, representation, using a synthesis windowingfunction having a meandering portion which passes a linear function incorrespondence of at least four points.

In has been noted that the analysis and/or synthesis windowing functionsused for the invention are particularly suitable for performing MDCT (orMDST) synthesis and analysis.

Examples of analysis/synthesis windowing functions for modulated lappedtransformation and methods and apparatus for using them are disclosedhereinafter.

BRIEF DESCRIPTION OF THE DRAWINGS

Embodiments of the present invention will be detailed subsequentlyreferring to the appended drawings, in which:

FIGS. 1 and 2 show schemes according to conventional technology.

FIGS. 3 to 12 show schemes of functions according to examples.

FIGS. 13 and 13A show encoders according to examples.

FIGS. 14 and 14A shows decoders according to examples.

FIGS. 15 and 16 show methods according to examples.

FIGS. 17 and 18 show apparatus according to examples.

FIG. 19 shows a scheme according to examples.

FIG. 20 shows a scheme according to conventional technology.

FIGS. 21 to 25 shows schemes according to examples.

FIGS. 26 to 31 show schemes of functions according to examples.

DETAILED DESCRIPTION OF THE INVENTION Examples

An information signal may be described in the time domain, TD, as asuccession of samples (e.g., xb(n) for a block b and index, e.g.,instant, n). The TD representation may be made of a plurality of frames,each associated to a plurality of samples. In the frequency domain, FD,a frame may be represented as a succession of bins (e.g., X (k), eachassociated to a particular frequency (each frequency being associated toan index k).

It is possible to convert a TD representation into an FD representationusing a modulated lapped transform technique (such as a modifieddiscrete cosine transform, MDCT, technique or a modified discrete sinetransform, MDST, technique, for example). Such a technique may beimplemented, for example, at an encoder side, so as to transform sampledvalues into frequency values.

It is possible to convert an FD representation into a TD representationusing an inverse modulated lapped transform technique (such as aninverse modified discrete cosine transform, IMDCT, technique or aninverse modified discrete sine, IMDST, technique). Such a technique maybe implemented, for example, at a decoder side, so as to transformfrequency values into TD values (e.g., for performing reproduction).

The table below indicates symbols used in the following sections.

Symbol Description N or N_(F) Frame size; number of sample processed inone frame f_(s) Sampling rate x_(b)(n) Time domain sample of block b andindex n X_(b)(k) Frequency domain bin of block b and frequency index k Zor L_(Z) Number of leading zeros in MDCT window

FIGS. 13 and 13A show encoder apparatus 130 and 130A, respectively. Eachof the encoder apparatus 130 or 130A may comprise a plurality of tools,such as functional blocks capable of performing signal processingtechniques. Each of these blocks may be a standalone device or may be afunctional block which divides a hardware structure with other tools ofthe encoder apparatus 130 and 130A.

The encoder apparatus 130 or 130A may comprise a modulated lappedtransform tool 131 (such as a low delay modified discrete cosinetransform, MDCT, tool, or a low delay modified discrete sine transform,MDST, tool, or another type of modulated lapped transform tool) whichmay convert an information signal (e.g., an audio signal) from a timedomain, TD, representation to a frequency domain, FD, representation.The modulated lapped transform tool 131 (e.g., MDCT or MDST tool)performs modulated lapped transform analysis (e.g., an MDCT analysis,MDST analysis).

Other tools may be provided. e.g., downstream to the modulated lappedtransform tool 131 or operating in TD. Some of them are here mentioned.

The encoder apparatus 130 may comprise a linear predictive coding, LPC,tool 132 for performing an LPC analysis in the FD. The encoder apparatus130A may comprise a spectral noise shaping, SNS, tool 132A forperforming an LPC analysis in the FD.

SNS may be seen as a generalization of LPC and, therefore, in someexamples, the LPC block 132 may be seen as a particular example of theSNS tool 132A (and the encoder apparatus 130 may be seen as a particularexample of the encoder apparatus 130A).

Each of the encoder apparatus 130 and 130A may comprise a temporal noiseshaping, TNS, tool 133, to control the temporal shape of noise withineach window of the information signal (e.g., as output by the modulatedlapped transform tool) in the FD.

Each of the encoder apparatus 130 and 130A may comprise a spectralquantizer 134 processing signals in the in the FD. The signal as outputby the TNS tool 133 may be quantized, e.g., using dead-zone plus uniformthresholds scalar quantization.

Each of the encoder apparatus 130 and 130A may comprise a noise levelestimator 136, e.g., downstream to the spectral quantizer 134.

Each of the encoder apparatus 130 and 130A may comprise a coder 135processing signals in the FD, for example, to perform entropy coding,e.g., to compress a bitstream. The coder 135 may, for example, performentropy coding.

Each of the encoder apparatus 130 and 130A may comprise a bandwidthdetector 137 a which may control, for example, a bandwidth controller atthe decoder.

Each of the encoder apparatus 130 and 130A may comprise tools whichprocess signals in the time domain, TD. For example, the encoderapparatus 130 or 130A may comprise a re-sampling tool 138 a (e.g., adownsampler) and/or a long term postfiltering, LTPF, tool 138 b, forcontrolling an LTPF active in TD at the decoder side.

Each of the encoder apparatus 130 and 130A may comprise a bitstreammultiplexer tool (bitstream writer) 137 to prepare a bitstream with dataobtained from TD and/or FD tools placed upstream.

The bitstream may comprise a digital representation of an informationsignal together with control data to be used at the decoder side. Thebitstream may be compressed or include portions which are compressed.

FIGS. 14 and 14A show decoder apparatus 140 and 140A, respectively. Eachof them may decode a digital representation of an information signal,e.g., as encoded by the encoder 130 or 130A, for example.

Each of the decoder apparatus 140 and 140A may comprise an inversemodulated lapped transform tool 147 (e.g., a low delay inverse MDCT toolor a low delay inverse DMST tool) to transform signal representationsfrom FD to TD. The modulated lapped transform tool 147 performs amodulated lapped transform synthesis (e.g., an MDCT synthesis and/or anMDST synthesis, etc.).

Other tools may be provided, e.g., upstream to the inverse modulatedlapped transform tool 147 or operating in TD. Some of them are herementioned.

Each of the decoder apparatus 140 and 140A may comprise a bitstreammultiplex tool 141 to obtain a bitstream (e.g., by transmission) from anencoder apparatus (e.g., the apparatus 130 or 130A). For example, anoutput from the encoder apparatus 130 or 130A may be provided as aninput signal to the decoder apparatus 140 or 140A.

Each of the decoder apparatus 140 and 140A may comprise a decoder 142which may, for example, decompress data in the bitstream. Arithmeticdecoding may be performed. A residual decoding may be performed.

Each of the decoder apparatus 140 and 140A may comprise a noise fillingtool 143 processing signals in the FD.

Each of the decoder apparatus 140 and 140A may comprise a global gaintool 144 processing signals in the FD.

Each of the decoder apparatus 140 and 140A may comprise a TNS decodertool 145 processing signals in the FD.

The decoder apparatus 140 may comprise an MDCT (or MDST) shaping tool146 (other lapped transformation tools are possible). The MDCT (or MDST)shaping tool 146 may process signals by applying gain factors computedfrom decoded LP filter coefficients (obtained from an LPC decoding tool146 a, for example) transformed to the FD spectrum (e.g., MDCT or MDSTspectrum, etc.).

The decoder apparatus 140A may comprise an SNS decoder tool 146A′, forexample, obtaining LPC coefficients from the SNS tool 132A.

Each of the decoder apparatus 140 and 140A may comprise an LTPF tool 148for performing a postfilter in the TD.

The decoder apparatus 140A may comprise a decoder bandwidth controller149 which may obtain bandwidth information from the bandwidth detector137 a, for example.

The encoder apparatus 130 (or 130A) and the decoder apparatus 140 (or140A) may be composed to each other to form a system.

Basically, the tools downstream to the modulated lapped transform tool131 in the encoder apparatus 130 or 130A and the tools upstream to theinverse modulated lapped transform tool 147 in the decoder apparatus 140or 140A may perform signal processing in the FD and are therefore FDtools.

Techniques are here discussed regarding the conversion techniques, e.g.,at the tools 131 and 147.

In general terms, the MDCT, MDST, etc., are discrete transforms whichhave the additional property of being lapped transforms. One of thesetransforms is performed on consecutive blocks of a larger dataset (e.g.,two consecutive frames). Subsequent blocks may be overlapped.Accordingly, a last part of one block may coincide with a first part ofa subsequent block. A windowing function w_(n)(n=0, . . . , 2N−1) may beused. In particular, the windowing function may be used as a weight. Anadvantage is that it is possible to avoid or reduce discontinuities atthe borders of the blocks.

FIG. 15 shows a method 150 for performing a modulated lapped transformanalysis, e.g., at the modulated lapped transform tool 131 (which may bea MDCT tool). The input signal x_(b)(n) of the current frame b mayconsist of N audio samples, where the newest one is located atx_(b)(N−1).

At step S151, input buffers may be updated. The time input buffer forthe modulated lapped transform t may be updated according to

t(n)=x _(b−1)(Z+n) for n=0 . . . N−1−Z

t(N−Z+n)=x _(b)(n) for n=0 . . . N−1

t(2N−Z+n)=0 for n=0 . . . Z−1

The latter is an initialization just used for consistency.

At step S152, a block of N time samples may be transformed to frequencycoefficients X(k) using the following equation:

${X(k)} = {\sqrt{\frac{2}{N}}{\sum\limits_{n = 0}^{{2N} - 1}{{{w_{N}(n)} \cdot {t(n)}}{\cos \left\lbrack {\frac{\pi}{N}\left( {n + \frac{1}{2} + \frac{N}{2}} \right)\left( {k + \frac{1}{2}} \right)} \right\rbrack}}}}$for  k = 0  …  N − 1

where w_(N) is an asymmetric modulated lapped transform window (e.g.,MDCT window) according to the used frame size.

FIG. 16 shows a method 160 for performing a modulated lapped transformsynthesis (e.g., MDCT synthesis), e.g., at the modulated lappedtransform tool 147 (which may be an IMDCT tool). Here, a spectrum

(k) (i.e., a representation in the FD) may be transformed to TD by thefollowing steps:

At step S161, a generation of time domain aliasing buffer

(n) of frame b may be performed.

For example:

${(n)} = {\sqrt{\frac{2}{N}}{\sum\limits_{k = 0}^{N - 1}{(k){\cos \left\lbrack {\frac{\pi}{N}\left( {n + \frac{1}{2} + \frac{N}{2}} \right)\left( {k + \frac{1}{2}} \right)} \right\rbrack}}}}$for  n = 0  …  2N − 1

At step S162, windowing of time domain aliasing buffer may be performed.For example:

(n)=w _(N)(2N−1−n)·

(n) for n=0 . . . 2N−1

At step S163, overlap-add operation to get reconstructed time samples

(n) of frame b may be conducted. For example:

(n)=

(N+Z+n)+

(Z+n) for n=0 . . . N−Z−1

(n)=

(Z+n) for n=N−Z . . . N−1

Notably, the windowing function W_(N)(n) at the analysis (analysiswindowing function) and the windowing function w_(N)(2N−1−n) at thesynthesis (synthesis windowing function) may be defined so as to be timereversed with each other. The window coefficients of the analysis andsynthesis window are time reversed versions of each other.

It has been noted that a particularly effective windowing function(e.g., synthesis windowing function) may be an analysis windowingfunction having a meandering portion crossing a linear function incorrespondence of four points.

FIG. 4 shows an analysis windowing function 40. The analysis windowingfunction 40 may be defined between 0 and 2N or between 0 and a valueintermediate between the 80% and the 85% or 90% of 2N (e.g., 81.25%). Insome examples, the analysis windowing function 40 may be defined in adifferent (e.g., broader) range.

The analysis windowing function 40 may be defined, at least for oneportion, with reference to a linear function 40′. In FIG. 4, the linearfunction 40′ is a constant function whose value is 1. However, thelinear function 40′ may be in general a linear function, e.g., definedin terms of y=an+b, where n is the index between the 0 and 2N (or itsnon-zero portion) and a and b are constants.

The analysis windowing function 40 may comprise a meandering portion 44.The meandering portion 44 may be so that the analysis windowing function40 encounters and/or crosses the linear function 40′ at four points #1,#2, #3, #4.

The meandering portion 44 may be so that the analysis windowing function40 is different from the linear function 40′ at the majority of indexese.g., for more than the 99% of the indexes.

The meandering portion 44 may be so that, at least at one indeximmediately preceding one of the indexes associated to the points(crossing points #1, #2, #3, #4, the value of the analysis windowingfunction 40 is greater than the value of the linear function (e.g., 1)at the crossing point (#1, #2, #3, of #4), and, at least at the indeximmediately subsequent the same index (associated to the crossingpoint), the value of the analysis windowing function 40 is smaller thanthe value of the linear function at the crossing point.

The meandering portion 44 may be so that, at least at one indeximmediately preceding one of the indexes associated to the crossingpoints #1, #2, #3, #4, the value of the analysis windowing function 40is smaller than the value of the linear function (e.g., 1) at thecrossing point (#1, #2, #3, of #4), and, at least at the indeximmediately subsequent the same index (associated to the crossingpoint), the value of the analysis windowing function 40 is greater thanthe value of the linear function at the crossing point.

The meandering portion 44 may be such that each of the crossing points#1, #2, #3, #4 is not immediately consecutive with the previous and thefollowing ones. For example, between two crossing points #1, #2, #3, and#4, there is at least one index of the analysis windowing function 40whose value does not coincide with the value of the analysis windowingfunction 40 at the same index.

The analysis windowing function 40 may be defined so as to be, incorrespondence to the meandering portion 44:

-   -   greater than the linear function 40′ in a first interval 41        between a first crossing point #1 and a second crossing point        #2;    -   lower than the linear function 40′ in a second interval 42        between the second crossing point #2 and a third crossing point        #3;    -   greater than the linear function 40′ in a third interval 43        between the third crossing point #3 and a fourth crossing point        #4.

The first crossing point #1 may precede the second crossing point #2,which may precede the third crossing point #3, which may precede thefourth crossing point #4.

The meandering portion 44 may extend between the first and the last(e.g., fourth) crossing point. The crossing points may be only four, insome examples.

The analysis windowing function 40 may also comprise an initialdecreasing portion 45, in which a negative minimum may be reached. Thenegative minimum may be localized between the 6.25% of 2N and the 18.75%of 2N. The analysis windowing function 40 may also comprise a rapidlyincreasing portion 46, rapidly increasing from the negative minimumtowards the meandering portion 44 (e.g., towards the first crossingpoint #1). The analysis windowing function 40 may also comprise arapidly decreasing portion 47, rapidly decreasing from the meanderingportion 44 to 0. The analysis windowing function 40 may also comprise aconstantly null portion 48, which may start between the 80% and the 85%or 90% of 2N (size of two frames) and continue to the end of 2N.

The absolute maximum value 41′ is in the first interval 41 (or in thethird interval).

The analysis windowing function 40 may, in the meandering portion 44,have a relative maximum value 43′ in the third interval 43 (or in thefirst interval) and a relative minimum value 42′ is in the secondinterval 42.

The analysis windowing function 40 may, in the meandering portion 44,have the values of the meandering window function 40 in correspondenceto the second interval 42 greater than 0.9. The second interval 42 maybe above 0.9.

The analysis windowing function 40 may present, in the meanderingportion 44, a value lower than the linear function 40′ in an intervalcomprised of the 30% and 50% of two frames (2N). In examples, after the50% of two frames (2N), the windowing function 40 may greater than thelinear function 40′. Hence, point #3 may be in correspondence to thelast index of the first frame or in correspondence to the first index ofthe second frame. In the interval 30% to 50% of two frames, themeandering window portion 44 may be below the linear function.

The maximum 41′ of the meandering window function 44 may be less than25% (and advantageously less than 5% or 3%) greater than the value ofthe linear function 40′ at the same index n.

The analysis windowing function 40 may present a first numericaldifferentiation between 0.02 and −0.02. The second numericaldifferentiation may be between −5*10⁻⁴ and +5*10⁻⁴, and advantageouslybetween −3*10⁻⁴ and +3*10⁻⁴. The third numerical differentiation may bebetween −2*10⁻⁵ and +2*10⁻⁵.

The linear function 40′ may be a non-increasing function. In theirexamples, the linear function 40′ may be a non-decreasing function. Thelinear function 40′ may have a value which is constant or varies ofmaximum +2% or −2%.

The analysis windowing function may be defined so as to be asymmetric.

In some examples, the analysis windowing function may be such that someof the interval (subsections) of the meandering portion 44 are inreverse order with respect to the description above.

FIGS. 26 and 27 show an analysis windowing function 40 (FIG. 27 is anenlarged view of a portion of FIG. 26). The analysis windowing function240 may be defined between 0 and 2N or between 0 and a valueintermediate between the 80% and the 85% or 90% of 2N (e.g., 81.25%). Insome examples, the analysis windowing function 40 may be defined in adifferent (e.g., broader) range.

The analysis windowing function 40 may be defined, at least for oneportion, with reference to a linear function 240′. In FIG. 27, thelinear function 240′ is a constant function whose value is 1. However,the linear function 40′ may be in general a linear function, e.g.,defined in terms of y=an+b, where n is the index between the 0 and 2N(or its non-zero portion) and a and b are constants (e.g., a=0, b=1).

The analysis windowing function 240 may comprise a meandering portion244. The meandering portion 244 may be so that the analysis windowingfunction 240 encounters and/or crosses the linear function 240′ at fourpoints #1, #2, #3, #4.

The meandering portion 244 may be so that the analysis windowingfunction 240 is different from the linear function 240′ at the majorityof indexes e.g., for more than the 99% of the indexes.

The meandering portion 244 may be so that, at least at one indeximmediately preceding one of the indexes associated to the points(crossing points) #1, #2, #3, #4, the value of the analysis windowingfunction 240 is greater than the value of the linear function 240′ atthe crossing point (#1, #2, #3, of #4), and, at least at the indeximmediately subsequent the same index (associated to the crossingpoint), the value of the analysis windowing function 240 is smaller thanthe value of the linear function 240′ at the crossing point (#1, #2, #3,of #4).

The meandering portion 244 may be so that, at least at one indeximmediately preceding one of the indexes associated to the crossingpoints #1, #2, #3, #4, the value of the analysis windowing function 40is smaller than the value of the linear function 240′ at the crossingpoint (#1, #2, #3, of #4), and, at least at the index immediatelysubsequent the same index (associated to the crossing point), the valueof the analysis windowing function 40 is greater than the value of thelinear function 240′ at the crossing point.

The meandering portion 244 may be such that each of the crossing points#1, #2, #3, #4 is not immediately consecutive with the previous and thefollowing ones. For example, between two crossing points #1, #2, #3, and#4, there is at least one index of the analysis windowing function 240whose value does not coincide with the value of the analysis windowingfunction 240 at the same index.

The analysis windowing function 240 may be defined so as to be, incorrespondence to the meandering portion 244:

-   -   greater than the linear function 240′ in a first interval 241        between a first crossing point #1 and a second crossing point        #2;    -   lower than the linear function 240′ in a second interval 242        between the second crossing point #2 and a third crossing point        #3;    -   greater than the linear function 240′ in a third interval 243        between the third crossing point #3 and a fourth crossing point        #4.

The first crossing point #1 may precede the second crossing point #2,which may precede the third crossing point #3, which may precede thefourth crossing point #4.

The meandering portion 244 may extend between the first and the last(e.g., fourth) crossing point. The crossing points may be only four, insome examples.

The analysis windowing function 240 may have all positive values. Theminimum may be a 0 value, which may be at the first index of theanalysis windowing function 240 and/or at the last indexes of theanalysis windowing function 240. The analysis windowing function 240 maycomprise a rapidly increasing portion 246, rapidly increasing towardsthe meandering portion 244 (e.g., the first crossing point #1). Theanalysis windowing function 240 may also comprise a rapidly decreasingportion 247, rapidly decreasing from the meandering portion 244 to 0.The analysis windowing function 240 may comprise a constantly nullportion 248, which may start between the 84% and the 90% of 2N (size oftwo frames) and continue to the end of 2N.

The absolute maximum value 241′ is in the first interval 41 (or in thethird interval).

The analysis windowing function 240 may, in the meandering portion 244,have a relative maximum value 243′ in the third interval 243 (or in thefirst interval) and/or a relative minimum value 242′ is in the secondinterval 242.

The analysis windowing function 240 may, in the meandering portion 244,have the values of the meandering window function 240 in correspondenceto the second interval 242 greater than 0.95. The second interval 242may be above 0.95. The first and/or third interval 241 and/or 243 may beless than 1.05.

The analysis windowing function 240 may present, in the meanderingportion 244, a value lower than the linear function 240′ in an intervalcomprised of the 30% and 50% of two frames (2N). In examples, after the50% of two frames (2N), the windowing function 240 may greater than thelinear function 240′. Hence, point #3 may be in correspondence to thelast index of the first frame or in correspondence to the first index ofthe second frame. In the interval 30% to 50% of two frames, themeandering window portion 244 may be below the linear function.

The maximum 241′ of the meandering window function 244 may be less than25% (and advantageously less than 5% or 3%) greater than the value ofthe linear function 240′ at the same index n.

The analysis windowing function 240 may present a first numericaldifferentiation between −0.01 and +0.01. The second numericaldifferentiation may be between −10⁻⁴ and +10⁻⁴. The third numericaldifferentiation may be between −10⁻⁵ and +10⁻⁻⁵.

The linear function 40′ may be a non-increasing function. In theirexamples, the linear function 40′ may be a non-decreasing function. Thelinear function 40′ may have a value which is constant or varies ofmaximum +2% or −2%.

The analysis windowing function may be defined so as to be asymmetric.

In some examples, the analysis windowing function 240 may be such thatsome of the interval (subsections) of the meandering portion 244 are inreverse order with respect to the description above.

FIGS. 28 and 29 compare the analysis windowing functions 40 and 240.

FIG. 6 shows another example of analysis windowing function 60. FIG. 7shows an enlarged view of the analysis windowing function 60 incorrespondence of a meandering portion 64.

In this case, the analysis windowing function 60 may be defined, atleast for one portion, with reference to a linear function 60′. In FIG.7, the linear function 60′ is a linear decreasing function (“diagonalline”).

The meandering portion 64 may be so that the analysis windowing function60 encounters and/or crosses the linear function 60′ at four points.

The analysis windowing function 40 may be defined so as to be, incorrespondence to the meandering portion 64:

-   -   greater than the linear function 60′ in a first interval 61′        between a first crossing point #1′ and a second crossing point        #2′;    -   lower than the linear function 60′ in a second interval 62′        between the second crossing #2′ point and a third crossing point        #3′;    -   greater than the linear function 60′ in a third interval 63′        between the third crossing point #3′ and a fourth crossing point        #4′;    -   lower than the linear function 60′ in a fourth interval 64′        between the fourth crossing point #4′ and a fifth crossing point        #5′;    -   greater than the linear function 60′ in a fifth interval 65′        between the fifth crossing point #5′ and a sixth crossing point        #6′.

The absolute maximum value may be in the first interval 61′ (in otherexamples, e.g., where the linear function is increasing, may be in thefifth interval).

The analysis windowing function 60 may, in the meandering portion 64,have a relative maximum value in the third interval 63′ and/or in thefifth interval 65′, a relative minimum value being in the secondinterval 62′ and/or in the fourth interval 64′.

The analysis windowing function 60 may, in the meandering portion 64,have the value of the windowing function in correspondence to at leastone of the first and third interval greater than 0.9 and in particular0.95 (in other examples, e.g., where the linear function is increasing,may be in the third and fifth interval).

The analysis windowing function 60 may present, in the meanderingportion 64, a value greater than the linear function 60′ in an intervalcomprised of the 30% and 50% of two frames (2N).

The maximum of the windowing function 60 may be less than 25% (andadvantageously less than 5%) greater than the value of the linearfunction 60 at the same index n.

The linear function 60′ may be a non-increasing function, in particulara strictly decreasing function.

The analysis windowing function may be defined so as to be asymmetric.

In examples, the analysis windowing function may have the intervalswhich are reversed with respect to FIG. 6.

In examples, a synthesis windowing function may be construed to besymmetric with respect to the analysis windowing function. For example,the symmetry axis may be in the middle of the 2N range (479 or 480, forexample).

An example of synthesis windowing function 90 is provided in FIG. 9(e.g., the synthesis windowing function associated to the analysiswindowing function 40 of FIG. 4). The synthesis windowing function 90may be defined between 0 and 2N or between a value intermediate betweenthe 10% or 15% and the 20% of 2N and 2N.

The synthesis windowing function 90 may be defined, at least for oneportion, with reference to a linear function (which in this case is notshown but is a constant value 1). In FIG. 9, the linear function may bea constant function whose value is 1. However, the linear function maybe in general a linear function, e.g., defined in terms of y=an+b.

The synthesis windowing function 90 may comprise a meandering portion94. The meandering portion 94 may be so that the synthesis windowingfunction 90 encounters and/or crosses the linear function 90′ (constant1, only partially shown for the sake of clarity) at four crossing points#1, #2, #3, #4 (only #2 being illustrated). The synthesis windowingfunction may cross the value 1 in the middle of the window (point #2),i.e. between sample N−1 and N.

The meandering portion 94 may be so that the synthesis windowingfunction 90 is different from the linear function 90′ at the majority ofindexes e.g., for more than the 99% of the indexes.

The meandering portion 94 may be so that, at least at one indeximmediately preceding one of the indexes associated to the points #1,#2, #3, #4, the value of the synthesis windowing function 90 is greaterthan the value of the linear function (e.g., 1) at the point (#1, #2,#3, of #4), and, at least at the index immediately subsequent the sameindex (associated to the point), the value of the synthesis windowingfunction 90 is smaller than the value of the linear function at thecrossing point. The meandering portion 94 may be so that, at least atone index immediately preceding one of the indexes associated to thepoints #1, #2, #3, #4, the value of the synthesis windowing function 90is smaller than the value of the linear function at the point (#1, #2,#3, of #4), and, at least at the index immediately subsequent the sameindex (associated to the crossing point), the value of the synthesiswindowing function 90 is greater than the value of the linear functionat the crossing point (#1, #2, #3, of #4).

The meandering portion 94 may be such that each of the points #1, #2,#3, #4 is not immediately consecutive with another of the points #1, #2,#3, #4. For example, between two points of #1, #2, #3, and #4, there maybe at least one index of the synthesis windowing function 90 whose valuedoes not coincide with the value of the synthesis windowing function 90at the same index.

The synthesis windowing function 90 may be defined so as to be, incorrespondence to the meandering portion 94:

-   -   greater than the linear function 90′ in a first interval 91        between a first crossing point #1 (not shown in FIG. 9, but        representable as symmetrical to point #4 in FIG. 4) and a second        crossing point #2 (which, in this case, is at the 480^(th)        sample, e.g., in the middle of the 2N range);    -   lower than the linear function 90′ in a second interval 92        between the second crossing point #2 and a third crossing point        #3 (not shown in FIG. 9, but representable as symmetrical to        point #3 in FIG. 4);    -   greater than the linear function 90′ in a third interval 93        between the third crossing point #3 and a fourth crossing point        #4 (not shown in FIG. 9, but representable as symmetrical to        point #1 in FIG. 4).

The synthesis windowing function 90 may also comprise a constantly nullportion 98, which may be located between the 10% and the 15% of 2N. Thesynthesis windowing function 90 may also comprise a rapidly increasingportion 97, rapidly increasing towards the meandering portion 94 from 0.The synthesis windowing function 90 may also comprise a rapidlydecreasing portion 96, rapidly decreasing towards a negative minimumfrom the meandering portion 94. The negative minimum may be localizedbetween the 81.25% of 2N and the 93.7% of 2N. The synthesis windowingfunction 90 may also comprise a final increasing portion 95, whichreaches 0 from the negative minimum.

The absolute maximum value may be in the third interval 93 (or in thefirst interval).

The synthesis windowing function 90 may, in the meandering portion 94,have a relative maximum value in the first interval 91 (or in the thirdinterval) and a relative minimum value is in the second interval.

The synthesis windowing function 90 may, in the meandering portion 94,have the value of the meandering window function in correspondence to atleast one of the first and third interval (91, 93) greater than 0.9. Thethird interval may be above 0.9.

The synthesis windowing function 90 may present, in the meanderingportion 94, a value greater than the linear function in an intervalcomprised of the 30% and 50% of two frames (2N).

The maximum of the synthesis windowing function 90 may be less than 25%(and advantageously less than 5%) greater than the value of the linearfunction at the same index n.

The synthesis windowing function 90 may present a numericaldifferentiation between 0.02 and −0.02. The second numericaldifferentiation may be between −5*10⁻⁴ and +5*10⁻⁴, and advantageouslybetween −3*10⁻⁴ and +3*10⁻⁴. The third numerical differentiation may bebetween −2*10⁻⁵ and +2*10⁻⁵.

The synthesis linear function 90 may be a non-increasing function. Inother examples, the synthesis linear function 90 may be a non-decreasingfunction. The synthesis linear function may 90 have a value which isconstant or varies of maximum +2% or −2%.

The synthesis windowing function 90 may be defined so as to beasymmetric.

In some examples, the synthesis windowing function 90 may be a functionsymmetrical with respect to the shape shown in FIG. 9. E.g., the valueat 0 would be the value which is at 960 in FIG. 9; the value at 1 wouldbe the value which is at 959 in FIG. 9, and so on.

An example of synthesis windowing function 290 is provided in FIG. 30(e.g., the synthesis windowing function associated to the analysiswindowing function 240 of FIGS. 26 to 29). The synthesis windowingfunction 290 may be defined between 0 and 2N or between a valueintermediate between the 10% or 15% and the 20% of 2N and 2N.

The synthesis windowing function 290 may be defined, at least for oneportion, with reference to a linear function 290′ (which in this case isnot shown but is a constant value 1).

In FIG. 30, the linear function 290′ may be a constant function whosevalue is 1. However, the linear function may be in general a linearfunction, e.g., defined in terms of y=an+b.

The synthesis windowing function 290 may comprise a meandering portion294. The meandering portion 294 may be so that the synthesis windowingfunction 290 encounters and/or crosses the linear function 290′(constant 1, only partially shown for the sake of clarity) at fourcrossing points #1, #2, #3, #4 (only #2 being illustrated). Thesynthesis windowing function may cross the value 1 in the middle of thewindow (point #2), i.e. between sample N−1 and N.

The meandering portion 294 may be so that the synthesis windowingfunction 290 is different from the linear function 290′ at the majorityof indexes e.g., for more than the 99% of the indexes.

The meandering portion 294 may be so that, at least at one indeximmediately preceding one of the indexes associated to the points #1,#2, #3, #4, the value of the synthesis windowing function 90 is greaterthan the value of the linear function (e.g., 1) at the point (#1, #2,#3, of #4), and, at least at the index immediately subsequent the sameindex (associated to the crossing point), the value of the synthesiswindowing function 90 is smaller than the value of the linear functionat the crossing point.

The meandering portion 294 may be so that, at least at one indeximmediately preceding one of the indexes associated to the points #1,#2, #3, #4, the value of the synthesis windowing function 290 is smallerthan the value of the linear function at the crossing point (#1, #2, #3,of #4), and, at least at the index immediately subsequent the same index(associated to the crossing point), the value of the synthesis windowingfunction 290 is greater than the value of the linear function at thecrossing point (#1, #2, #3, of #4).

The meandering portion 294 may be such that each of the crossing points#1, #2, #3, #4 is not immediately consecutive with another of the points#1, #2, #3, #4. For example, between two crossing points of #1, #2, #3,and #4, there may be at least one index of the synthesis windowingfunction 290 whose value does not coincide with the value of thesynthesis windowing function 290 at the same index.

The synthesis windowing function 290 may be defined so as to be, incorrespondence to the meandering portion 294.

-   -   greater than the linear function 290′ (e.g., greater than 1) in        a first interval 291 between a first crossing point #1 and a        second crossing point #2 (which, in this case, is at the        480^(th) sample, e.g., in the middle of the 2N range);    -   lower than the linear function 290′ in a second interval 292        between the second crossing point #2 and a third crossing point        #3;    -   greater than the linear function 290′ in a third interval 293        between the third crossing point #3 and a fourth crossing point        #4.

The synthesis windowing function 290 may comprise a constantly nullportion 298, which may be located. at one border of the window (2Nsamples) and may comprise between 10% and 15% of the samples. Thesynthesis windowing function 290 may also comprise a rapidly increasingportion 297, rapidly increasing towards the meandering portion 294 from0. The synthesis windowing function 290 may comprise a rapidlydecreasing portion 296, rapidly decreasing towards 0 from the meanderingportion 294.

The absolute maximum value may be in the third interval 293 (or in thefirst interval).

The synthesis windowing function 290 may, in the meandering portion 294,have a relative maximum value in the first interval 291 (or in the thirdinterval) and a relative minimum value is in the second interval.

The synthesis windowing function 290 may, in the meandering portion 294,have the value of the meandering window function in correspondence to atleast one of the first and third interval (291, 293) greater than 0.9 or0.95. The third interval may be above 0.95.

The synthesis windowing function 290 may present, in the meanderingportion 294, a value greater than the linear function in an intervalcomprised of the 30% and 50% of two frames (2N).

The maximum of the synthesis windowing function 290 may be less than 25%(and advantageously less than 5% or 3%) greater than the value of thelinear function at the same index n.

The synthesis windowing function 290 may present a numericaldifferentiation between −0.01 and +0.01. The second numericaldifferentiation may be between −10⁻⁴ and +10⁻⁴. The third numericaldifferentiation may be between −10⁻⁵ and +10⁻⁻⁵.

The synthesis linear function 90 may be a non-increasing function. Inother examples, the synthesis linear function 90 may be a non-decreasingfunction. The synthesis linear function may 90 have a value which isconstant or varies of maximum +2% or −2%.

The synthesis windowing function 90 may be defined so as to beasymmetric.

In some examples, the synthesis windowing function 90 may be a functionsymmetrical with respect to the shape shown in FIG. 9. E.g., the valueat 0 would be the value which is at 960 in FIG. 9; the value at 1 wouldbe the value which is at 959 in FIG. 9, and so on.

Discussion

It has been noted that this function has particularly interestingfeatures which are extremely suited for the MDCT (or MDST or other) andIMDCT (or IMDST or other) techniques, in particular for audio coding.

As a metric for the smoothness of MDCT window shape, we use thenumerical differentiation like

${{dw}(t)} = {{\frac{{w\left( {1 + h} \right)} - {w\left( {t - h} \right)}}{2h}\mspace{14mu} {for}\mspace{14mu} 1} = {{h\mspace{14mu} \ldots \mspace{14mu} 2N} - 1 - h}}$

Another metric in order to assess the quality of MDCT window is temporalshape of the quantization error which is described by

S _(q)(t)=w ²(t+N)+w ²(t) for t=0 . . . N−1

The quantization error may be introduced in the spectral domain as partof an audio coding process.

The ALDO window is here discussed. ALDO is a design approach forasymmetric window providing an improved temporal shape. The designprocess of the ALDO window as used in the EVS codec is described in(3GPP) and [2]. FIG. 1 shows a schematic window shape for a ALDO windowscheme.

An ALDO window consists of four sections, a raising function, a sectionof strict ones, a decay function and zeros. The number of zeros in w₄also determines the number of ones in w₂, with w₂=2*w₄. How to obtain w₁and w₃ is described in [1] and [2].

FIG. 2 shows an ALDO window designed according to [1] and [2] forparameters Lz=180 (number of leading zeroes in the MDCT window, W₄(n) inFIG. 1) and the total window length is 960 samples.

As can be seen in the fourth subplot of FIG. 2 (“Time modulationquantization error”), the ALDO shows a good temporal shape of thequantization error which mean the energy sum of two subsequentoverlapping synthesis windows is close to one. The optimal result wouldinvolve only ones which are achieved by symmetric window shapes.

The drawbacks of this design approach are mainly visible in the secondsubplot of FIG. 2 (“Numerical differentiation”), where the numericaldifferentiation is outlined. It can be seen that the window shape is anot a continuously differentiable mathematical function and is thereforenon-optimal regarding the frequency response. Due to static ones in thewindow shape, smooth transitions coming from the raise function andgoing to the decay function are not possible.

The present invention breaks up with the design constraint to maintain asection of strict ones in order to optimize the temporal shape. Instead,the ones are replaced by a sequence of values meandering around a linearfunction (e.g., a constant value, such as “1”). This solution allows thesmooth and continuous window shape while keeping the temporal shapeclose to an optimal state.

FIG. 3 shows an example window designed with a meandering section(portion) according to the invention. As can be seen in the fourthsubplot of FIG. 3 (“Time modulation quantization error”), the window'sproperty regarding the temporal shaping of the quantization error issimilar to the ALDO window. However, comparing the second subplot ofFIG. 3 (“Numerical differentiation”) to second subplot of FIG. 2 (ALDOwindow), it may be understood that the invention permits to generatewindow shapes which are continuously differentiable. This enables forinstance the design of window shapes with higher stop band attenuation,so far not possible with design process used for the ALDO window.

FIG. 4 outlines an example window shape 40 in more detail. In thisexample, the meandering section is based on crossing, in correspondenceof four points, a linear function (in this case, a constant value “1”).Even a higher number of crossings is possible but four is considered asthe minimum number.

Such a window can be the result of a mathematical optimization processcombining one error function for the frequency response with and errorfunction representing the deviation from the optimal temporal shaping ofthe quantization noise.

Some implementation involved specific data formats, e.g. fix-pointimplementations. For such implementation, the value range of the windowcoefficients might be scales as values above one involve a differentscaling. In such cases, the meandering section might not be around one.Only after descaling, the values get into the obvious range.

FIG. 5 shows an example how the MDCT window coefficients 50 arerepresented inside an fix-point implementation. It is an examplerepresentation where some values are scaled by 2^(x) and other by2^(x−1) due to fix point architecture.

Considering a general case, the window does not necessarily meanderaround 1 but may meander around a virtual diagonal line without crossingthe one. FIG. 6 shows three different windows, i.e.

-   -   w₁: New optimized meandering around 1 (windowing function 70);    -   w₂: G.718 window (not meandering at all) (windowing function        70′);    -   w₃: New optimized window for special case: Meandering around a        diagonal (windowing function 60).

The described MDCT window fulfils the perfect reconstruction property.For that, we consider the MDCT to be decomposed as a windowingoperation, a time domain alias cancellation (TDAC) step and a discretecosine transform (DCT) type IV kernel, as for instance described in [3].

Therefore, the windowing and TDAC can be seen as a folding step onanalysis and an unfolding step on synthesis side. At the latter one, theunfolded sequences of two blocks are combined in the overlap-and-addoperation. See FIG. 8, which refers to a scheme of a block 1 (81) and ablock 2 (82).

For the overlapping region on the analysis side, the windowing and TDACof block 1 can be described as

P ₁(n)=w _(a)(N+n)−w _(a)(2N−1−n) for n=0 . . . N−1

-   -   and block 2

P ₂(n)=w _(a)(N+n)+w _(a)(N−1−n) for n=0 . . . N−1

For the syntheses side, P1 and P2 are folded-out, windowed andoverlap-and-add is performed:

O(n)=w _(s)(N+n)P ₁(n)+w _(s)(n)P ₂(n) for n=0 . . . N−1

This can be written as

O(n)=w _(a)(N+n)w _(s)(N+n)−w _(a)(2N−1−n)w _(s)(N+n)+w _(a)(n)w_(s)(n)+w _(a)(N−1−n)w _(s)(n)

-   -   for n=0 . . . N−1

which can be separated into two components:

-   -   PR1: Perfect reconstruction component

w _(a)(N+n)w _(s)(N+n)+w _(a)(n)w _(s)(n)=1

-   -   for n=0 . . . N−1    -   PR2: Alias cancellation component

w _(a)(N−1−n)w _(s)(n)−w _(a)(2N−1−n)w _(s)(N+n)=0

-   -   for n=0 . . . N−1

FIGS. 9-12 show analysis and synthesis sequences.

Reference is now made to FIGS. 19 and 20 with reference to the numericaldifferentiation of the present analysis and synthesis windowingfunctions.

By considering also the 2^(nd) and 3^(rd) numerical differentiation, itis possible to clearly distinguish the properties of present examples(e.g., FIG. 19 with reference to the example of FIG. 4, and FIG. 31 withreference to the example of FIG. 26) with those of conventionaltechnology (ALDO, FIG. 20).

As a metric for the smoothness of modulated lapped transform windowshape (e.g., MDCT or MDST window shape), we may use the numericaldifferentiation like:

${{dw}(t)} = {{\frac{{w\left( {t + h} \right)} - {w\left( {t - h} \right)}}{2h}\mspace{14mu} {for}\mspace{14mu} t} = {{h\mspace{14mu} \ldots \mspace{14mu} 2N} - 1 - h}}$

In order to weight the importance of each coefficient, i.e. higheramplitudes have a higher weight, the following describes a weighted.

${{wdw}(t)} = {{{{dw}(t)}\frac{{w\left( {t + h} \right)} + {w\left( {t - h} \right)}}{2}\mspace{14mu} {for}\mspace{14mu} t} = {{h\mspace{14mu} \ldots \mspace{14mu} 2N} - 1 - h}}$

This allows neglecting low level window coefficients in the analysis ofthe continuity of the window shape.

In FIGS. 19 and 20 and 31, the 1^(st) numerical differentiation isplotted in the regular and weighted version (dashed). 1902 and 3102refer to the 1^(st) numerical differentiation for the inventivewindowing function; 1904 and 3104 (dashed) refer to the weighted versionof the 1^(st) numerical differentiation for the inventive windowingfunction. 2002 refers to the 1^(st) numerical differentiation for theALDO windowing function; 2004 (dashed) refers to the weighted version ofthe 1^(st) numerical differentiation for the ALDO windowing function.

For the 2^(nd) and 3^(rd) numerical differentiation (ND), the weightedversion of the 1^(st) one is used. 1910 and 3110 refer to the weightedversion of the 2^(nd) numerical differentiation for the inventivewindowing function. 2010 refers to the weighted version of the 2^(nd)numerical differentiation for the ALDO windowing function. 1920 refersto the weighted version of the 3^(rd) numerical differentiation for theinventive windowing function. 2020 and 3120 refer to the weightedversion of the 3^(rd) numerical differentiation for the ALDO windowingfunction.

As can be observed, the 2^(nd) ND version shows some clear peaks 2021for the ALDO window while the inventive windows show only moderatechanges. For the 3^(rd) ND, also the amplitude level of both windowsclear differentiate and can be expressed.

An analysis of the degree of freedom for window design for ALDO and forthe inventive window function is here presented.

The ALDO design approach [2] offers only the parameters C1, C2 and thenumber of zeros Lz as degree of freedom. Any focus regarding near fieldor far field attenuation is not possible by the proposed method as thegiven degree of freedom in the design process is very limited.

In order to extend the ALDO algorithm and allowing a higher degree offreedom, the design approach in [2] has to be given up and has to beexchanged by a numerical optimization method. In a first step thesection of strict ones in the middle of the window is maintained whilethe free window coefficients (or design parameter) are optimized.

Assuming that analysis and synthesis window are typically time reversedversions of each other, the perfect reconstruction constraint can beexpressed by

w _(a)(N+n)w _(s)(N+n)+w _(a)(n)w _(s)(n)=1 for n=0 . . . N−1

→w(N+n)w(2N−n)+w(n)w(N+n)=1 for n=0 . . . N−1

This can typically be fulfilled by restricting the number of freecoefficients to 1.5N, e.g. by

w(2N−n)=(1−w(n)w(N+n))/w(N+n) for n=0 . . . N−1

The zero section is located inside w(2N−n). For the constraint of thestrict ones in the middle, 2*Lz of the window coefficients areconsidered as non-free coefficients. For a transformation length ofN=480 and a window size of 2N and Lz=180, this leads to a degree offreedom of 1.5N−2*Lz=360 free coefficients. This window is referred toas intermediate window.

The new invention increases the degree of freedom to the maximum byexchanging the portion of strict ones by a meandering section (e.g., 44,64, 94) around one or around any other linear (e.g., diagonal) line(e.g., 40′, 60′, 90′). Thus, only half of the middle section needs to beexcluded from the free coefficients. Consequently, the number ofavailable coefficients is increased to 1.5*480−180=540 freecoefficients. This allows to maximally optimize the frequency responsetowards any indented focus.

FIGS. 21 and 22 show a comparison of the design approaches. Inparticular, FIG. 21 shows a comparison window shape ALDO (211) vsintermediate (212) vs new invention (213). FIG. 22 shows a zoom(enlarged view) of window shapes ALDO (211), intermediate (212), newinvention (213).

The intermediate window and the new invention window are optimizedtowards the same focus.

FIG. 23 shows a plot frequency response with ALDO (231), intermediate(232), new invention (233). The intermediate (231) and new inventionwindow (233) can focus on different areas, achieving in this case abetter far field attenuation. This may be used for instance to avoid anyaliasing distortions caused by band limited signals or whenever signalsare resampled through a filter bank. This focus is impossible for theALDO window, as there are only two parameters available controlling theshape. Comparing the intermediate and the new invention window, itbecomes evident that the new invention achieves much less fluctuationsin the response due to the very continuous shape in the time domain.

FIG. 24 shows a zoom (enlarged view) of frequency response comparingintermediate (242) and new invention (243). As can be observed, the newinvention window (243) typically shows a significant higher attenuationof all side lopes. This confirms that the meandering portion (44, 64,94) is beneficial in the window design over a portion of strict ones.

With reference of the time modulation of window, it is noted that themain motivation for the ALDO window was to combine an asymmetric shapewith a maximum amplitude of 1, in order to avoid large temporalfluctuations of the quantization error. The quantization error is addedin the frequency domain and its temporal shape is controlled by thesynthesis window shape.

Actually, here also the meandering sequence even improves this windowproperty. FIG. 25 shows a direct comparison the time modulation of thequantization error caused by an ALDO window (253) and a proposedinvention (251). As can be seen, the maximum temporal distortion islower for the new invention and therefore provides less noticeabledistortions.

In examples above, the meandering portion may be, for example, incorrespondence (at least partial) with the DCT kernel. In examples, atleast one first crossing point (e.g., #1) is in the first frame, whileat least another crossing point (e.g., #3) is in the subsequent frame.In examples, one crossing point (e.g., #2) at the border between thefirst and the subsequent frames.

In examples, “meandering” refers to the fact that there are at leastfour intersections with a linear function (the meandering portioncrosses the linear function in at least two or in some cases in at leastfour points). For example, a meandering portion may have an increasingpart, followed by a decreasing part, followed by an increasing part, andso on, or a decreasing part, followed by an increasing part, followed bya decreasing part, and so on.

Other Examples

FIG. 17 shows an apparatus 110 which may implement the encoder apparatus130 (or 130A) and/or MDCT tool 131 and/or perform at least some steps ofthe method 150. The apparatus 110 may comprise a processor 111 and anon-transitory memory unit 112 with a block 114 storing instructionswhich, when executed by the processor 111, may cause the processor 111to perform a MDCT synthesis. The non-transitory memory unit 112 may alsostore, block 113, an analysis windowing function, e.g., as discussedabove. The apparatus 110 may comprise an input unit 116, which mayobtain an input information signal (e.g., an audio signal). A transitorystorage space 118 may be reserved for saving data to be updated (e.g.,input buffers).

FIG. 18 shows an apparatus 120 which may implement the decoder apparatus140 (or 140A) and/or the IMDCT 147 and/or perform the method 160. Theapparatus 120 may comprise a processor 121 and a non-transitory memoryunit 122 with a block 124 storing instructions which, when executed bythe processor 121, may cause the processor 121 to perform, inter alia,an MDCT synthesis process. The apparatus 120 may comprise an input unit126, which may obtain a decoded representation of an information signal(e.g., an audio signal) in the FD. The processor 121 may thereforeperform processes to obtain a TD representation of the informationsignal. This decoded representation may be provided to external unitsusing an output unit 127. The output unit 127 may comprise, for example,a communication unit to communicate to external devices (e.g., usingwireless communication, such as Bluetooth) and/or external storagespaces. The processor 121 may save the decoded representation of theaudio signal in a local storage space 128.

In examples, the apparatus 110 and 120 may be the same device. Inexamples, the apparatus 110 and 120 exchange information signals and/orcontrol data e.g., wirelessly, e.g., using protocol Bluetooth.

Depending on certain implementation requirements, examples may beimplemented in hardware. The implementation may be performed using adigital storage medium, for example a floppy disk, a Digital VersatileDisc (DVD), a Blu-Ray Disc, a Compact Disc (CD), a Read-only Memory(ROM), a Programmable Read-only Memory (PROM), an Erasable andProgrammable Read-only Memory (EPROM), an Electrically ErasableProgrammable Read-Only Memory (EEPROM) or a flash memory, havingelectronically readable control signals stored thereon, which cooperate(or are capable of cooperating) with a programmable computer system suchthat the respective method is performed. Therefore, the digital storagemedium may be computer readable.

Generally, examples may be implemented as a computer program productwith program instructions, the program instructions being operative forperforming one of the methods when the computer program product runs ona computer. The program instructions may for example be stored on amachine readable medium.

Other examples comprise the computer program for performing one of themethods described herein, stored on a machine-readable carrier. In otherwords, an example of method is, therefore, a computer program having aprogram instruction for performing one of the methods described herein,when the computer program runs on a computer.

A further example of the methods is, therefore, a data carrier medium(or a digital storage medium, or a computer-readable medium) comprising,recorded thereon, the computer program for performing one of the methodsdescribed herein. The data carrier medium, the digital storage medium orthe recorded medium are tangible and/or non-transitionary, rather thansignals which are intangible and transitory.

A further example comprises a processing unit, for example a computer,or a programmable logic device performing one of the methods describedherein.

A further example comprises a computer having installed thereon thecomputer program for performing one of the methods described herein.

A further example comprises an apparatus or a system transferring (forexample, electronically or optically) a computer program for performingone of the methods described herein to a receiver. The receiver may, forexample, be a computer, a mobile device, a memory device or the like.The apparatus or system may, for example, comprise a file server fortransferring the computer program to the receiver.

In some examples, a programmable logic device (for example, a fieldprogrammable gate array) may be used to perform some or all of thefunctionalities of the methods described herein. In some examples, afield programmable gate array may cooperate with a microprocessor inorder to perform one of the methods described herein. Generally, themethods may be performed by any appropriate hardware apparatus.

The above described examples are illustrative for the principlesdiscussed above. It is understood that modifications and variations ofthe arrangements and the details described herein will be apparent. Itis the intent, therefore, to be limited by the scope of the impendingpatent claims and not by the specific details presented by way ofdescription and explanation of the examples herein.

Some Examples of Windowing Functions

Numerical examples are here provided. As no mathematical formula inclosed form for obtaining the function has been found, examples areherewith provided. The values hereinbelow may be provided in backword orforward order, in the sense that in some examples the first value isassociated to the time instant n=0 and the last value is associated tothe last time instant n=2N−1 (forward direction), while in some examplesthe first value is associated to the time instant n=2N−1 and the lastvalue is associated to the first time instant n=0 (backward direction).In some cases, the analysis windowing function and the synthesiswindowing function are taken from the same list, but the analysiswindowing function is read in the backward (or forward) direction, whilethe synthesis windowing function is read in the forward (or backward)direction.

In order to embody at least one of the examples below, it is possible toextract a sub-succession of at least 10 values (e.g., consecutivevalues), when the 10 values are different from 0 (e.g., all the samplesor at least the majority thereof may form the at least 10 values). Atolerance or ±1% may be possible; in some cases ±0.5%, ±0.05%, in othercases, ±2%, ±5%, in other cases 0%.

Examples Associated to FIGS. 4 and 9

In the examples, a notation such as −7.078546706512391e−04f means−7.078546706512391*10⁻⁴. “f” refers to the notation in floating point(in some cases, it may be omitted).

A numerical example of windowing function w₃₀, for frame size N=80, isherewith provided by the 160 entries for n=0 . . . 2N−1. As explainedabove, the last values may be constantly 0.

−7.078546706512391e−04f, −2.098197727900724e−03f,−4.525198076002370e−03f, −8.233976327300612e−03f,−1.337713096257934e−02f, −1.999721557401502e−02f,−2.800909464274782e−02f, −3.721502082245055e−02f,−4.731768261606175e−02f, −5.794654834034055e−02f,−6.867606753531441e−02f, −7.904647440788692e−02f,−8.859705468085925e−02f, −9.688303623049199e−02f,−1.034961241263523e−01f, −1.080766457616878e−01f,−1.103242262600913e−01f, −1.099809851424550e−01f,−1.068172142230882e−01f, −1.006190418791648e−01f,−9.116452506492527e−02f, −7.820617483254730e−02f,−6.146688124166948e−02f, −4.063362855701623e−02f,−1.536329520788766e−02f, +1.470155068746303e−02f,+4.989736509080558e−02f, +9.050369257152079e−02f,+1.366911019414417e−01f, +1.884686389218322e−01f,+2.456456803467095e−01f, +3.077789078889820e−01f,+3.741642373060188e−01f, +4.438114799213576e−01f,+5.154735456539700e−01f, +5.876661722564289e−01f,+6.587619767809000e−01f, +7.270576699841359e−01f,+7.908752989295335e−01f, +8.486643364959733e−01f,+8.991320235484349e−01f, +9.413348145272842e−01f,+9.747634827941575e−01f, +9.994114730415857e−01f,+1.015760373791603e+00f, +1.024736164069697e+00f,+1.027634294456205e+00f, +1.025991493983836e+00f,+1.021427210603284e+00f, +1.015439859549357e+00f,+1.009366925499550e+00f, +1.003508162416449e+00f,+9.988898206257559e−01f, +9.953133902427869e−01f,+9.925943919208190e−01f, +9.905771957917731e−01f,+9.891371616557014e−01f, +9.881790747212391e−01f,+9.876249269174586e−01f, +9.874056275509585e−01f,+9.874524849192456e−01f, +9.876951134084213e−01f,+9.880640617030884e−01f, +9.884926873551375e−01f,+9.889230031022089e−01f, +9.893074965384659e−01f,+9.896146331889107e−01f, +9.898319269347060e−01f,+9.899693102025342e−01f, +9.900603352632121e−01f,+9.901575015155720e−01f, +9.903255289051605e−01f,+9.906303787150326e−01f, +9.911298894709990e−01f,+9.918665491182922e−01f, +9.928619727154252e−01f,+9.941156069136238e−01f, +9.956033775539884e−01f,+9.972793109558521e−01f, +9.990784840729244e−01f,+1.000922365901945e+00f, +1.002728111386909e+00f,+1.004416038098237e+00f, +1.005919224127911e+00f,+1.007189345025525e+00f, +1.008200146369426e+00f,+1.008949493525753e+00f, +1.009458241425143e+00f,+1.009768980817384e+00f, +1.009940336228694e+00f,+1.010039453539107e+00f, +1.010132323996401e+00f,+1.010272524848519e+00f, +1.010494354532353e+00f,+1.010808068774316e+00f, +1.011201071127927e+00f,+1.011641272406023e+00f, +1.012080125934687e+00f,+1.012458183122033e+00f, +1.012706955800289e+00f,+1.012755013843985e+00f, +1.012530134411619e+00f,+1.011962331100864e+00f, +1.010982135506986e+00f,+1.009512438049510e+00f, +1.007460860286395e+00f,+1.004708677491086e+00f, +1.001111413242302e+00f,+9.965041017623596e−01f, +9.907199995730845e−01f,+9.823765865983288e−01f, +9.708821747608998e−01f,+9.546732976073705e−01f, +9.321553861564006e−01f,+9.018003682081348e−01f, +8.623984077953557e−01f,+8.132817365236141e−01f, +7.544551974836834e−01f,+6.866580716267418e−01f, +6.113488038789190e−01f,+5.306181649316597e−01f, +4.471309850999502e−01f,+3.639114681156236e−01f, +2.841647033392408e−01f,+2.110209448747969e−01f, +1.472287968327703e−01f,+9.482665349502291e−02f, +5.482436608328477e−02f,+2.701461405056264e−02f, +9.996743588367519e−03f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f

A numerical example of windowing function w₁₆₀, for frame size N=160, isherewith provided by the 320 entries for n=0 . . . 2N−1. As explainedabove, the last values may be constantly 0.

−4.619898752628163e−04f, −9.747166718929050e−04f,−1.664473096973725e−03f, −2.597106916737789e−03f,−3.806285163352241e−03f, −5.324608721716763e−03f,−7.175885277771099e−03f, −9.382480860899108e−03f,−1.195270300743193e−02f, −1.489528159506296e−02f,−1.820666399965468e−02f, −2.187570925786862e−02f,−2.588471937157619e−02f, −3.020862738245264e−02f,−3.481597793538342e−02f, −3.967067992672979e−02f,−4.472698045914417e−02f, −4.994225863256500e−02f,−5.526334794593565e−02f, −6.063717235243996e−02f,−6.600961519440657e−02f, −7.131966266443390e−02f,−7.651178225890490e−02f, −8.152964005319532e−02f,−8.631137544905677e−02f, −9.080411291245728e−02f,−9.495377758870335e−02f, −9.870736514214426e−02f,−1.020202684361974e−01f, −1.048438825017798e−01f,−1.071382314127799e−01f, −1.088690135027248e−01f,−1.099969655786929e−01f, −1.104898474883336e−01f,−1.103225838568563e−01f, −1.094621746650760e−01f,−1.078834293141886e−01f, −1.055612509762041e−01f,−1.024650162703341e−01f, −9.857014566194629e−02f,−9.384684920715425e−02f, −8.826309993000785e−02f,−8.178792716809512e−02f, −7.438785600211463e−02f,−6.602189797715241e−02f, −5.665655641133161e−02f,−4.624456893420224e−02f, −3.474585776145929e−02f,−2.211581608120528e−02f, −8.310425696208936e−03f,+6.717697635290676e−03f, +2.300642061077823e−02f,+4.060106462625085e−02f, +5.953239090915557e−02f,+7.983354189816511e−02f, +1.015233140203748e−01f,+1.246171387327525e−01f, +1.491152519299797e−01f,+1.750067399059861e−01f, +2.022699854906251e−01f,+2.308655379767671e−01f, +2.607365124918583e−01f,+2.918144694729168e−01f, +3.240095704645023e−01f,+3.572175180786021e−01f, +3.913146885756875e−01f,+4.261571642320424e−01f, +4.615925445090212e−01f,+4.974471592901086e−01f, +5.335326819631583e−01f,+5.696546730080154e−01f, +6.056083823929643e−01f,+6.411830842823245e−01f, +6.761653499550255e−01f,+7.103400549562944e−01f, +7.434943718765665e−01f,+7.754281892901473e−01f, +8.059437233154637e−01f,+8.348589373399948e−01f, +8.620108336276733e−01f,+8.872599706865123e−01f, +9.104863121445679e−01f,+9.315962496426278e−01f, +9.505220861927248e−01f,+9.672366712325431e−01f, +9.817397501303696e−01f,+9.940557180662704e−01f, +1.004247514102417e+00f,+1.012407428282884e+00f, +1.018650990561848e+00f,+1.023118841384460e+00f, +1.025972450969440e+00f,+1.027397523939210e+00f, +1.027585830688143e+00f,+1.026738673647482e+00f, +1.025061777648234e+00f,+1.022756514615106e+00f, +1.020009139549275e+00f,+1.016996499560845e+00f, +1.013915946100629e+00f,+1.011044869639164e+00f, +1.007773858455400e+00f,+1.004848753962734e+00f, +1.002245009135684e+00f,+9.999393169239009e−01f, +9.979055415627330e−01f,+9.961203379971326e−01f, +9.945597525471822e−01f,+9.932031606606762e−01f, +9.920297273323891e−01f,+9.910230654424902e−01f, +9.901668953434221e−01f,+9.894488374513719e−01f, +9.888556356037892e−01f,+9.883778520531268e−01f, +9.880051626345804e−01f,+9.877295459610343e−01f, +9.875412739766566e−01f,+9.874329809802893e−01f, +9.873949921033299e−01f,+9.874197049003676e−01f, +9.874973205882319e−01f,+9.876201238703241e−01f, +9.877781920433015e−01f,+9.879637979933339e−01f, +9.881678007807095e−01f,+9.883835200189653e−01f, +9.886022219397892e−01f,+9.888182771263505e−01f, +9.890247977602895e−01f,+9.892178658748239e−01f, +9.893923680007577e−01f,+9.895463342815009e−01f, +9.896772011542693e−01f,+9.897859195209235e−01f, +9.898725363809847e−01f,+9.899410789223559e−01f, +9.899945557067980e−01f,+9.900394023736973e−01f, +9.900814722948890e−01f,+9.901293790312005e−01f, +9.901902265696609e−01f,+9.902734448815004e−01f, +9.903862280081246e−01f,+9.905379830873822e−01f, +9.907348826312993e−01f,+9.909842592301273e−01f, +9.912905118607647e−01f,+9.916586940166509e−01f, +9.920906151219310e−01f,+9.925887208794144e−01f, +9.931516528513824e−01f,+9.937790866568735e−01f, +9.944668184371617e−01f,+9.952116634297566e−01f, +9.960068616185641e−01f,+9.968461329825753e−01f, +9.977203369515556e−01f,+9.986213520769593e−01f, +9.995382582242990e−01f,+1.000461955079660e+00f, +1.001380551217109e+00f,+1.002284871786226e+00f, +1.003163845364970e+00f,+1.004009147462043e+00f, +1.004811375053364e+00f,+1.005563968008037e+00f, +1.006259855360867e+00f,+1.006895570408563e+00f, +1.007466616298057e+00f,+1.007972441990187e+00f, +1.008411468616852e+00f,+1.008786009787269e+00f, +1.009097763850333e+00f,+1.009351762546296e+00f, +1.009552401900961e+00f,+1.009707093778162e+00f, +1.009822090220407e+00f,+1.009906958448099e+00f, +1.009969021400474e+00f,+1.010017890428877e+00f, +1.010060809299530e+00f,+1.010106564965965e+00f, +1.010161131093372e+00f,+1.010231078494249e+00f, +1.010319484524512e+00f,+1.010430470494512e+00f, +1.010564099281000e+00f,+1.010721360243234e+00f, +1.010899655674578e+00f,+1.011096993993037e+00f, +1.011308167670753e+00f,+1.011529185153809e+00f, +1.011753008569803e+00f,+1.011973876511603e+00f, +1.012182837094955e+00f,+1.012373028737774e+00f, +1.012535058602453e+00f,+1.012660975529858e+00f, +1.012740575296603e+00f,+1.012765922449960e+00f, +1.012726958954961e+00f,+1.012615904116265e+00f, +1.012422888521601e+00f,+1.012140460211194e+00f, +1.011758810583150e+00f,+1.011269960947744e+00f, +1.010663676735228e+00f,+1.009930754807923e+00f, +1.009058249873833e+00f,+1.008034308295421e+00f, +1.006843352506855e+00f,+1.005470005637052e+00f, +1.003894772403371e+00f,+1.002098854400575e+00f, +1.000060686758758e+00f,+9.977600196406868e−01f, +9.951746430061121e−01f,+9.922861082472264e−01f, +9.890757868707590e−01f,+9.847362453480265e−01f, +9.798613526271561e−01f,+9.741378617337759e−01f, +9.673331975559332e−01f,+9.592539757044516e−01f, +9.496984081652284e−01f,+9.384634163826711e−01f, +9.253567968750328e−01f,+9.101986790930605e−01f, +8.928338316495705e−01f,+8.731437835983047e−01f, +8.510420440685049e−01f,+8.264839911291133e−01f, +7.994681492797084e−01f,+7.700431275216928e−01f, +7.383028603058783e−01f,+7.043814340356083e−01f, +6.684616478236647e−01f,+6.307755329382612e−01f, +5.915799587176216e−01f,+5.511703155400274e−01f, +5.098915423728179e−01f,+4.681017110047964e−01f, +4.261772971493010e−01f,+3.845172335531009e−01f, +3.435228672445613e−01f,+3.036004651973099e−01f, +2.651434678028531e−01f,+2.285283969438072e−01f, +1.941021906320984e−01f,+1.621735416384830e−01f, +1.330015240938615e−01f,+1.067840430193724e−01f, +8.365057236623041e−02f,+6.365188111381356e−02f, +4.676538412257621e−02f,+3.288072750732215e−02f, +2.183057564646270e−02f,+1.336381425803019e−02f, +6.758124889697787e−03f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f.

A numerical example of windowing function w₂₄₀, for frame size N=240, isherewith provided (480 samples):

−3.613496418928369e−04f, −7.078546706512391e−04f,−1.074443637110903e−03f, −1.533478537964509e−03f,−2.098197727900724e−03f, −2.778420871815740e−03f,−3.584129920673041e−03f, −4.525198076002370e−03f,−5.609327243712055e−03f, −6.843234536105624e−03f,−8.233976327300612e−03f, −9.785314755557023e−03f,−1.149880303071551e−02f, −1.337713096257934e−02f,−1.542181679511618e−02f, −1.762979910961727e−02f,−1.999721557401502e−02f, −2.252080561390149e−02f,−2.519406300389030e−02f, −2.800909464274782e−02f,−3.095765092956728e−02f, −3.402996266948349e−02f,−3.721502082245055e−02f, −4.050053247568393e−02f,−4.387219218706189e−02f, −4.731768261606175e−02f,−5.082325342672667e−02f, −5.437166635159518e−02f,−5.794654834034055e−02f, −6.153426201732499e−02f,−6.511708163113709e−02f, −6.867606753531441e−02f,−7.219447805250771e−02f, −7.565695975592170e−02f,−7.904647440788692e−02f, −8.234442557322251e−02f,−8.553324579905185e−02f, −8.859705468085925e−02f,−9.152091100798199e−02f, −9.428847446755965e−02f,−9.688303623049198e−02f, −9.929123258537813e−02f,−1.015008467688577e−01f, −1.034961241263523e−01f,−1.052637003544443e−01f, −1.067939984687745e−01f,−1.080766457616878e−01f, −1.090997300590506e−01f,−1.098524491515805e−01f, −1.103242262600913e−01f,−1.105084619148789e−01f, −1.103977408741932e−01f,−1.099809851424550e−01f, −1.092492774392824e−01f,−1.081974227416502e−01f, −1.068172142230882e−01f,−1.050995803285455e−01f, −1.030360111111103e−01f,−1.006190418791648e−01f, −9.784120023411771e−02f,−9.469304216883027e−02f, −9.116452506492527e−02f,−8.724644532866996e−02f, −8.293043914044632e−02f,−7.820617483254730e−02f, −7.306142427456862e−02f,−6.748468182105991e−02f, −6.146688124166948e−02f,−5.499497258200362e−02f, −4.805444424454820e−02f,−4.063362855701623e−02f, −3.272045590229335e−02f,−2.430122582451853e−02f, −1.536329520788766e−02f,−5.891434269890659e−03f, +4.126595858583295e−03f,+1.470155068746303e−02f, +2.584738191459814e−02f,+3.757652772246801e−02f, 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+0.000000000000000e+00f}

A numerical example of windowing function w₃₂₀, for frame size N=320, isherewith provided (640 samples):

−3.021153494057143e−04f, −5.867737487939294e−04f,−8.366504004139796e−04f, −1.126635355725494e−03f,−1.470492941694331e−03f, −1.873473391018495e−03f,−2.339292362082021e−03f, −2.872008069419264e−03f,−3.476256385086407e−03f, −4.155963816705528e−03f,−4.914563787665504e−03f, −5.755172503953251e−03f,−6.680623380533122e−03f, −7.693816924650567e−03f,−8.796760749750191e−03f, −9.990503073705982e−03f,−1.127574117138621e−02f, −1.265334152129685e−02f,−1.412438986522702e−02f, −1.568889620430290e−02f,−1.734512089366117e−02f, −1.909097368362797e−02f,−2.092546711168754e−02f, −2.284684792818856e−02f,−2.485207716234951e−02f, −2.693746704328349e−02f,−2.909952486193999e−02f, −3.133504629493832e−02f,−3.363960728361352e−02f, −3.600820974457969e−02f,−3.843601741746971e−02f, −4.091746034850161e−02f,−4.344654894948344e−02f, −4.601786724624048e−02f,−4.862598509282497e−02f, −5.126474204655663e−02f,−5.392644753556616e−02f, −5.660384311081047e−02f,−5.929116747072080e−02f, 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+0.000000000000000e+00f

A numerical example of windowing function w₄₈₀, for frame size N=480, isherewith provided (960 samples):

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−2.873390902713615e−02f,−3.020862738245264e−02f, −3.171440372994384e−02f,−3.325098858986303e−02f, −3.481597793538342e−02f,−3.640892406933019e−02f, −3.802742318209150e−02f,−3.967067992672979e−02f, −4.133575417353826e−02f,−4.302203371734278e−02f, −4.472698045914417e−02f,−4.645022292934329e−02f, −4.818891490266687e−02f,−4.994225863256500e−02f, −5.170690802826666e−02f,−5.348162036097223e−02f, −5.526334794593565e−02f,−5.705123152423822e−02f, −5.884271749745559e−02f,−6.063717235243996e−02f, −6.243104027829089e−02f,−6.422303545004304e−02f, −6.600961519440657e−02f,−6.778962269634495e−02f, −6.955996868581379e−02f,−7.131966266443390e−02f, −7.306581273272733e−02f,−7.479758913001458e−02f, −7.651178225890490e−02f,−7.820711420768856e−02f, −7.988010693411644e−02f,−8.152964005319532e−02f, −8.315237353264004e−02f,−8.474728946770714e−02f, −8.631137544905677e−02f,−8.784374452959058e−02f, −8.934164364321417e−02f,−9.080411291245728e−02f, −9.222795761428432e−02f,−9.361232867223340e−02f, 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+0.000000000000000e+00f

A numerical example of windowing function w₉₆₀, for frame size N=960, isherewith provided (1920 samples):

−1.596869453315999e−04, −3.021153494057143e−04, −4.142323860121641e−04,−5.058484031439142e−04, −5.867737487939294e−04, −6.666034929771656e−04,−7.499813122143762e−04, −8.366504004139794e−04, −9.273096237893370e−04,−1.023773491532728e−03, −1.126635355725494e−03, −1.235247794937702e−03,−1.349627853510606e−03, −1.470492941694331e−03, −1.598145399394582e−03,−1.732450700747454e−03, −1.873473391018495e−03, −2.021493133113876e−03,−2.176721793926887e−03, −2.339292362082021e−03, −2.509298051958086e−03,−2.686801873698675e−03, −2.872008069419264e−03, −3.065254982136409e−03,−3.266686736002951e−03, −3.476256385086407e−03, −3.694123506258334e−03,−3.920651988943251e−03, −4.155963816705528e−03, −4.399940204176431e−03,−4.652692827862344e−03, −4.914563787665504e−03, −5.185653529531935e−03,−5.465821495567872e−03, −5.755172503953251e−03, −6.054056765330685e−03,−6.362583863449497e−03, −6.680623380533122e−03, −7.008304674718039e−03,−7.346013465075499e−03, −7.693816924650567e−03, −8.051465262788033e−03,−8.419001443263380e−03, −8.796760749750191e−03, −9.184751507972289e−03,−9.582657230262597e−03, −9.990503073705982e−03, −1.040865597419619e−02,−1.083716113754551e−02, −1.127574117138621e−02, −1.172444049589271e−02,−1.218362542421523e−02, −1.265334152129685e−02, −1.313331245904876e−02,−1.362355747138776e−02, −1.412438986522702e−02, −1.463576680723352e−02,−1.515729094197161e−02, −1.568889620430290e−02, −1.623084814367250e−02,−1.678306524600559e−02, −1.734512089366117e−02, −1.791696516251699e−02,−1.849892453107246e−02, −1.909097368362797e−02, −1.969273308274951e−02,−2.030414376031160e−02, −2.092546711168754e−02, −2.155660557689242e−02,−2.219710359529686e−02, −2.284684792818856e−02, −2.350606779320675e−02,−2.417463749934085e−02, −2.485207716234951e−02, −2.553826580898371e−02,−2.623344061522424e−02, −2.693746704328349e−02, −2.764983997782559e−02,−2.837043858580717e−02, −2.909952486193998e−02, −2.983695676902939e−02,−3.058218787751172e−02, −3.133504629493832e−02, 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+0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00.

Examples Associated to FIGS. 26 to 30

A numerical example of windowing function w₄₀, for frame size N=40, isherewith provided by the 80 entries for n=0 . . . 2N−1. As explainedabove, the last values may be zeros.

+9.959086585790517e−04, +3.819056787237678e−03, +9.540832613229890e−03,+1.921659800166160e−02, +3.382719081038548e−02, +5.424831667522354e−02,+8.120777668775610e−02, +1.152171887125930e−01, +1.564942331034909e−01,+2.049363422022628e−01, +2.601166575816199e−01, +3.212814164616093e−01,+3.873472997948746e−01, +4.569497078592333e−01, +5.285192958868393e−01,+6.003522489375573e−01, +6.706896380227332e−01, +7.378044458510402e−01,+8.000925313431716e−01, +8.561409184410547e−01, +9.048272294524792e−01,+9.453685031730190e−01, +9.773507430600533e−01, +1.000800872826561e+00,+1.016171590112097e+00, +1.024315247630982e+00, +1.026415431432931e+00,+1.023858366571912e+00, +1.018135705524407e+00, +1.010794822557756e+00,+1.003406509762925e+00, +9.967831265986109e−01, +9.920995520917141e−01,+9.892206942816891e−01, +9.879658322200813e−01, +9.881273531631907e−01,+9.894805541465801e−01, +9.917849916000535e−01, +9.947847580943504e−01,+9.982119669301160e−01, +1.001791235858836e+00, +1.005242583245485e+00,+1.008283053756130e+00, +1.010631281038659e+00, +1.012015300253356e+00,+1.012180753005270e+00, +1.010896765282633e+00, +1.007963362035220e+00,+1.003227255072391e+00, +9.966050551498514e−01, +9.868284225039941e−01,+9.731250287581631e−01, +9.540636479502398e−01, +9.283864275822276e−01,+8.950916858157935e−01, +8.534769362643825e−01, +8.032090930429980e−01,+7.444735201251689e−01, +6.780787033699449e−01, +6.053970453856138e−01,+5.282077505750667e−01, +4.486552956056635e−01, +3.691875990296312e−01,+2.924566408966777e−01, +2.210718537110463e−01, +1.573148583944309e−01,+1.030525757797768e−01, +5.982732244758054e−02, +2.871831923385133e−02,+9.683884928956490e−03, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00.

A numerical example of windowing function w₈₀, for frame size N=80, isherewith provided by the 160 entries for n=0 . . . 2N−1. As explainedabove, the last values may be zeros.

+6.143388180964179e−04, +1.489582832987000e−03, +2.884104959764029e−03,+4.934298832466617e−03, +7.779130464154915e−03, +1.154910606525086e−02,+1.637155619860352e−02, +2.237116158648752e−02, +2.966159685753317e−02,+3.835663329277230e−02, +4.855610986150206e−02, +6.035055738891727e−02,+7.382288203064732e−02, +8.903563687211119e−02, +1.060356225286319e−01,+1.248534855777947e−01, +1.454931890869180e−01, +1.679435556337752e−01,+1.921728622634411e−01, +2.181238261985594e−01, +2.457259744642953e−01,+2.748839432649996e−01, +3.054824712370942e−01, +3.373873799614014e−01,+3.704415932452488e−01, +4.044749630814483e−01, +4.393004362003260e−01,+4.747225454237193e−01, +5.105341492548225e−01, +5.465201916422433e−01,+5.824658100332457e−01, +6.181452662624718e−01, +6.533411462740817e−01,+6.878367295965062e−01, +7.214176027060971e−01, +7.538887973483771e−01,+7.850546571907628e−01, +8.147397447696774e−01, +8.427819363777799e−01,+8.690376742017057e−01, +8.933935477349644e−01, +9.157483563218768e−01,+9.360270196617569e−01, +9.541731142261065e−01, +9.701635474343885e−01,+9.840036439809510e−01, +9.957199420334376e−01, +1.005374268639838e+00,+1.013046655758663e+00, +1.018843380560658e+00, +1.022896948293643e+00,+1.025355286710874e+00, +1.026382881625701e+00, +1.026155530733488e+00,+1.024853974580724e+00, +1.022664602721801e+00, +1.019779396547454e+00,+1.016391686789653e+00, +1.012697033320358e+00, +1.008885191761748e+00,+1.005378742804807e+00, +1.001563778373068e+00, +9.982531564931281e−01,+9.954346644968789e−01, +9.930950268060122e−01, +9.912170911359961e−01,+9.897805192546195e−01, +9.887624937408933e−01, +9.881383235740961e−01,+9.878819413827574e−01, +9.879662130250981e−01, +9.883630508181326e−01,+9.890434070785485e−01, +9.899772316163624e−01, +9.911334564321237e−01,+9.924800441092685e−01, +9.939841207305906e−01, +9.956121471675398e−01,+9.973300590248015e−01, +9.991033633647473e−01, +1.000897441314013e+00,+1.002677088643863e+00, +1.004407190937699e+00, +1.006052289109999e+00,+1.007576934100958e+00, +1.008945862447015e+00, +1.010124241309341e+00,+1.011077969726137e+00, +1.011773962181442e+00, +1.012180362866919e+00,+1.012266707295288e+00, +1.012004064757857e+00, +1.011365223023975e+00,+1.010324996851905e+00, +1.008860731864438e+00, +1.006952983357691e+00,+1.004586273379809e+00, +1.001749900308864e+00, +9.984386632116344e−01,+9.946500332901397e−01, +9.895756853352172e−01, +9.838303127859196e−01,+9.769999155793757e−01, +9.689141159310996e−01, +9.594038121639412e−01,+9.483086322505029e−01, +9.354860218216989e−01, +9.208101305030523e−01,+9.041732260327581e−01, +8.854882249661838e−01, +8.646864947605046e−01,+8.417237467711145e−01, +8.165875713256009e−01, +7.892986353718001e−01,+7.599171886893816e−01, +7.285474515411827e−01, +6.953282935906302e−01,+6.604334017809461e−01, +6.240661431421666e−01, +5.864461424698465e−01,+5.478160663871147e−01, +5.084499758302218e−01, +4.686361426418982e−01,+4.286789889246253e−01, +3.889032719013045e−01, +3.496431418636314e−01,+3.112360816586544e−01, +2.740128472224535e−01, +2.382847225401666e−01,+2.043379825955252e−01, +1.724305860483632e−01, +1.427939789949265e−01,+1.156385879569741e−01, +9.115821766571995e−02, +6.952749039054593e−02,+5.088975408628225e−02, +3.533430192568954e−02, +2.286680405144430e−02,+1.338005016725895e−02, +6.640506529168652e−03, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00.

A numerical example of windowing function w₁₂₀, for frame size N=120, isherewith provided by the 240 entries for n=0 . . . 2N−1. As explainedabove, the last values may be zeros.

+5.087227626168386e−04, +9.959086585790517e−04, +1.682208006328800e−03,+2.609697259047744e−03, +3.819056787237678e−03, +5.349319592933909e−03,+7.243906383895192e−03, +9.540832613229890e−03, +1.227637642543709e−02,+1.548950238899404e−02, +1.921659800166160e−02, +2.349369619441617e−02,+2.835199581667961e−02, +3.382719081038548e−02, +3.994939538719628e−02,+4.674775238543380e−02, +5.424831667522354e−02, +6.247770776443612e−02,+7.145835917501348e−02, +8.120777668775610e−02, +9.174400412319896e−02,+1.030764959637497e−01, +1.152171887125930e−01, +1.281665713944242e−01,+1.419264381068653e−01, +1.564942331034909e−01, +1.718593189799504e−01,+1.880134254543744e−01, +2.049363422022628e−01, +2.226123055761096e−01,+2.410151242797736e−01, +2.601166575816199e−01, +2.798871008989962e−01,+3.002880135563586e−01, +3.212814164616093e−01, +3.428208463088390e−01,+3.648596557863134e−01, +3.873472997948746e−01, +4.102294951869188e−01,+4.334494534591082e−01, +4.569497078592333e−01, 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+1.006315717067918e+00,+1.007332693127034e+00, +1.008283053756130e+00, +1.009156423082384e+00,+1.009942535308151e+00, +1.010631281038659e+00, +1.011212744622770e+00,+1.011677230257499e+00, +1.012015300253356e+00, +1.012217779097186e+00,+1.012275790821109e+00, +1.012180753005270e+00, +1.011924425888915e+00,+1.011498917644724e+00, +1.010896765282633e+00, +1.010110965619444e+00,+1.009135094671655e+00, +1.007963362035220e+00, +1.006590756505588e+00,+1.005013115379014e+00, +1.003227255072391e+00, +1.001231060075500e+00,+9.990235555436858e−01, +9.966050551498514e−01, +9.939894706113089e−01,+9.904539200261149e−01, +9.868284225039941e−01, +9.827716736909488e−01,+9.782206672373213e−01, +9.731250287581631e−01, +9.674323528812744e−01,+9.610947043524248e−01, +9.540636479502398e−01, +9.462952991190324e−01,+9.377489107516087e−01, +9.283864275822276e−01, +9.181762606422500e−01,+9.070861558801854e−01, +8.950916858157935e−01, +8.821696237804294e−01,+8.683025287048570e−01, +8.534769362643825e−01, +8.376852006833730e−01,+8.209275259764013e−01, +8.032090930429980e−01, +7.845450482523652e−01,+7.649554851899686e−01, +7.444735201251689e−01, +7.231348066419057e−01,+7.009860555207412e−01, +6.780787033699450e−01, +6.544686506489734e−01,+6.302212149502727e−01, +6.053970453856138e−01, +5.800715766089168e−01,+5.543129276657669e−01, +5.282077505750727e−01, +5.018369724442092e−01,+4.752902962082383e−01, +4.486552956056652e−01, +4.220281118338883e−01,+3.955057965950340e−01, +3.691875990296320e−01, +3.431732847389720e−01,+3.175633015043183e−01, +2.924566408966782e−01, +2.679463783886042e−01,+2.441231331518492e−01, +2.210718537110466e−01, +1.988719153219592e−01,+1.775967625327044e−01, +1.573148583944310e−01, +1.380903364946733e−01,+1.199837497591550e−01, +1.030525757797769e−01, +8.735085011789188e−02,+7.292811584897502e−02, +5.982732244758056e−02, +4.808178837444506e−02,+3.771135297837851e−02, +2.871831923385135e−02, +2.108352028641225e−02,+1.476289412849005e−02, +9.683884928956495e−03, +5.642168789286858e−03,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00.

A numerical example of windowing function w₁₆₀, for frame size N=160, isherewith provided by the 320 entries for n=0 . . . 2N−1. As explainedabove, the last values may be 0.

+4.595886345493055e−04, +7.919323614002698e−04, +1.227927169310031e−03,+1.783653266717233e−03, +2.479549413444207e−03, +3.329799454594261e−03,+4.353535478916468e−03, +5.564965156664018e−03, +6.986108359341676e−03,+8.629882322202329e−03, +1.051343406844975e−02, +1.265082642578719e−02,+1.506090447446532e−02, +1.775591229287213e−02, +2.075475983187825e−02,+2.406813715401559e−02, +2.771207863541604e−02, +3.169933248543932e−02,+3.604609640533871e−02, +4.076128638095439e−02, +4.586038120884381e−02,+5.135136676471998e−02, +5.724780220726930e−02, +6.355854744461048e−02,+7.029450733434550e−02, +7.745987198268531e−02, +8.506635369887924e−02,+9.311641620512773e−02, +1.016162955027316e−01, +1.105690806271684e−01,+1.199789286645804e−01, +1.298417294090302e−01, +1.401623800497866e−01,+1.509371564593891e−01, +1.621632295622287e−01, +1.738354123649302e−01,+1.859520359191026e−01, +1.985008828937603e−01, +2.114778554475382e−01,+2.248732557074316e−01, +2.386763947872762e−01, +2.528729453658238e−01,+2.674547009618951e−01, +2.824031465430401e−01, +2.977050145264297e−01,+3.133419120661713e−01, +3.292976696294886e−01, +3.455490160824131e−01,+3.620795045342974e−01, +3.788648665671841e−01, +3.958851576591690e−01,+4.131143794748322e−01, +4.305308301005456e−01, +4.481076715576617e−01,+4.658227790464821e−01, +4.836466393241829e−01, +5.015564851667653e−01,+5.195228071176610e−01, +5.375197039843709e−01, +5.555183841040963e−01,+5.734957812557457e−01, +5.914186654649489e−01, +6.092622887527459e−01,+6.269981160888640e−01, +6.446002007776794e−01, +6.620384583071039e−01,+6.792906550106088e−01, +6.963256426589250e−01, +7.131194393772130e−01,+7.296469905863920e−01, +7.458864594794676e−01, +7.618094719403713e−01,+7.773958448163656e−01, +7.926208751337592e−01, +8.074666387233143e−01,+8.219101564897180e−01, +8.359343163788637e−01, +8.495180470826319e−01,+8.626485837105826e−01, +8.753083234662220e−01, +8.874884715160425e−01,+8.991737724042251e−01, +9.103527429187326e−01, +9.210144133066616e−01,+9.311556192776946e−01, +9.407644740241826e−01, +9.498382236872068e−01,+9.583732599601223e−01, +9.663690412284377e−01, +9.738235617865406e−01,+9.807442506043361e−01, +9.871297972052695e−01, +9.929872268444632e−01,+9.983241398929388e−01, +1.003150760219063e+00, +1.007473713377193e+00,+1.011309151636166e+00, +1.014666681083198e+00, +1.017563337333301e+00,+1.020014681326785e+00, +1.022039872150903e+00, +1.023654257342442e+00,+1.024881624147540e+00, +1.025739288978437e+00, +1.026250709375593e+00,+1.026436666375082e+00, +1.026320857404224e+00, +1.025922917798664e+00,+1.025269979527211e+00, +1.024382188798244e+00, +1.023284940887058e+00,+1.022000829220643e+00, +1.020555973231408e+00, +1.018971390778550e+00,+1.017275179369116e+00, +1.015489129111694e+00, +1.013639356938881e+00,+1.011747750709711e+00, +1.009840844244693e+00, +1.007939764480188e+00,+1.006407400915498e+00, +1.004374825095777e+00, +1.002469814737132e+00,+1.000689073754539e+00, +9.990346001249977e−01, +9.975024904153303e−01,+9.960941547576162e−01, +9.948051243621099e−01, +9.936362728142866e−01,+9.925826537087717e−01, +9.916447007525191e−01, +9.908170758245324e−01,+9.900998445795673e−01, +9.894873685512386e−01, +9.889794323427195e−01,+9.885701787626714e−01, +9.882591911282058e−01, +9.880404423341358e−01,+9.879133688360181e−01, +9.878718098237022e−01, +9.879150762106034e−01,+9.880368938846610e−01, +9.882364564839506e−01, +9.885073687439192e−01,+9.888487088987707e−01, +9.892539488627546e−01, +9.897220412447528e−01,+9.902463287269285e−01, +9.908256340476208e−01, +9.914531811725067e−01,+9.921276814881759e−01, +9.928422499725458e−01, +9.935955098307742e−01,+9.943804814776256e−01, +9.951957244449919e−01, +9.960341878404958e−01,+9.968943831870675e−01, +9.977692009836100e−01, +9.986571134591464e−01,+9.995509738170480e−01, +1.000449227898040e+00, +1.001344692310058e+00,+1.002235786606954e+00, +1.003115291715261e+00, +1.003981602446902e+00,+1.004827468041713e+00, +1.005651275972376e+00, +1.006445772052972e+00,+1.007209352772459e+00, +1.007934783656087e+00, +1.008620496650569e+00,+1.009259314290145e+00, +1.009849742422788e+00, +1.010384692193296e+00,+1.010862783160582e+00, +1.011277044709547e+00, +1.011626247430694e+00,+1.011903571699736e+00, +1.012107954864219e+00, +1.012232755709885e+00,+1.012277089047072e+00, +1.012234505114778e+00, +1.012104319978655e+00,+1.011880293122688e+00, +1.011561972516341e+00, +1.011143373963981e+00,+1.010624321020038e+00, +1.009999148545101e+00, +1.009268031808824e+00,+1.008425698479647e+00, +1.007472774447058e+00, +1.006404483571931e+00,+1.005222003295591e+00, +1.003921160689206e+00, +1.002503762756151e+00,+1.000966332772540e+00, +9.993114007411373e−01, +9.975362702189898e−01,+9.956442306333592e−01, +9.936333924912825e−01, +9.908677480361242e−01,+9.882326326262749e−01, +9.853620567056602e−01, +9.822305093671991e−01,+9.788185853162172e−01, +9.751026333215268e−01, +9.710631852370086e−01,+9.666759668947944e−01, +9.619242192293307e−01, +9.567841986369235e−01,+9.512394303101863e−01, +9.452700238623795e−01, +9.388615698236068e−01,+9.319946435581106e−01, +9.246592033932568e−01, +9.168383396399868e−01,+9.085218034087421e−01, +8.996967011299613e−01, +8.903562054918268e−01,+8.804877931535187e−01, +8.700884209228057e−01, +8.591492134848259e−01,+8.476686394755906e−01, +8.356428970797861e−01, +8.230753889817990e−01,+8.099649296155544e−01, +7.963204506324437e−01, +7.821460539775005e−01,+7.674541821769616e−01, +7.522563457568547e−01, +7.365702052057368e−01,+7.204090552899627e−01, +7.037975107157410e−01, +6.867542812151157e−01,+6.693041888771051e−01, +6.514710959179395e−01, +6.332854832820911e−01,+6.147685389896460e−01, +5.959553778639692e−01, +5.768737955463938e−01,+5.575534287167304e−01, +5.380320138068979e−01, +5.183454027643563e−01,+4.985259415650634e−01, +4.786156067459849e−01, +4.586473038370941e−01,+4.386643656872842e−01, +4.187046888325280e−01, +3.988123056192917e−01,+3.790262923635886e−01, +3.593914828698096e−01, +3.399474132903109e−01,+3.207392420889753e−01, +3.018061113177048e−01, +2.831905952786929e−01,+2.649288369241889e−01, +2.470608550624402e−01, +2.296201119084317e−01,+2.126433716126151e−01, +1.961601816145380e−01, +1.802035203864437e−01,+1.647996883470626e−01, +1.499787548077656e−01, +1.357643522991611e−01,+1.221842534547464e−01, +1.092601994264172e−01, +9.701788451015501e−02,+8.547680283183663e−02, +7.465976378295235e−02, +6.458254322751883e−02,+5.526281189874138e−02, +4.670976978373095e−02, +3.893244425578719e−02,+3.192976013776996e−02, +2.569810636390756e−02, +2.022259265088492e−02,+1.548317776486452e−02, +1.144924909653903e−02, +8.076482660383199e−03,+5.300044080947794e−03, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00.

A numerical example of windowing function w₂₄₀, for frame size N=240, isherewith provided (480 samples):

+4.090106504820579e−04, +6.143388180964179e−04, +8.571759876954877e−04,+1.147015057857495e−03, +1.489582832987000e−03, +1.889770382231583e−03,+2.353000800169909e−03, +2.884104959764029e−03, +3.488213786635855e−03,+4.170040431489613e−03, +4.934298832466617e−03, +5.787076505403503e−03,+6.733811743137561e−03, +7.779130464154915e−03, +8.927044958757816e−03,+1.018202888968871e−02, +1.154910606525086e−02, +1.303349217699797e−02,+1.463951288465963e−02, +1.637155619860352e−02, +1.823455383898077e−02,+2.023309488998589e−02, +2.237116158648752e−02, +2.465237348403478e−02,+2.708101935270475e−02, +2.966159685753317e−02, +3.239884850877327e−02,+3.529601774976465e−02, +3.835663329277230e−02, +4.158447932459513e−02,+4.498322421745353e−02, +4.855610986150206e−02, +5.230596475016741e−02,+5.623624576084146e−02, +6.035055738891727e−02, +6.465186317477950e−02,+6.914195749790462e−02, +7.382288203064732e−02, +7.869709331995660e−02,+8.376761638427657e−02, +8.903563687211118e−02, +9.450199243028472e−02,+1.001680193006426e−01, +1.060356225286319e−01, +1.121060220821844e−01,+1.183788547045326e−01, +1.248534855777947e−01, +1.315302847610869e−01,+1.384103079528939e−01, +1.454931890869180e−01, +1.527772946853750e−01,+1.602608842337125e−01, +1.679435556337752e−01, +1.758245615079801e−01,+1.839020119821303e−01, +1.921728622634411e−01, +2.006344295681524e−01,+2.092853879977170e−01, +2.181238261985594e−01, +2.271462264407930e−01,+2.363479205237173e−01, +2.457259744642953e−01, +2.552771551741124e−01,+2.649981094228982e−01, +2.748839432649996e−01, +2.849296444030153e−01,+2.951306505827265e−01, +3.054824712370942e−01, +3.159799638238941e−01,+3.266169794538543e−01, +3.373873799614014e−01, +3.482855915638277e−01,+3.593057693987583e−01, +3.704415932452488e−01, +3.816862385160692e−01,+3.930329782788047e−01, +4.044749630814483e−01, +4.160051103939122e−01,+4.276159595085598e−01, +4.393004362003260e−01, +4.510516333434032e−01,+4.628616046119925e−01, +4.747225454237193e−01, 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A numerical example of windowing function w₃₂₀, for frame size N=320, isherewith provided (640 samples). As explained above, the last values maybe zeros.

+3.821992968116373e−04, +5.337292962876158e−04, +7.010318350640769e−04,+8.910357994632938e−04, +1.107726421735994e−03, +1.354305556720234e−03,+1.632968062524617e−03, +1.944392065549996e−03, +2.291251920978034e−03,+2.676357471188558e−03, +3.102160324883627e−03, +3.569302917352426e−03,+4.080147004222224e−03, +4.637409387087253e−03, +5.243697651112320e−03,+5.900394401532485e−03, +6.609908124535202e−03, +7.375089966000295e−03,+8.197610411215884e−03, +9.078195201482999e−03, +1.001888876231112e−02,+1.102272421949215e−02, +1.209201783148431e−02, +1.322773320467288e−02,+1.443170227861997e−02, +1.570673344965518e−02, +1.705481769961258e−02,+1.847711280171765e−02, +1.997546609433320e−02, +2.155255505248360e−02,+2.320993080187466e−02, +2.494800748775296e−02, +2.676898100961685e−02,+2.867550388692567e−02, +3.066976865911597e−02, +3.275241094528637e−02,+3.492474656851959e−02, +3.718932683936013e−02, +3.954748420753697e−02,+4.200013716597904e−02, +4.454866443036003e−02, 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A numerical example of windowing function w₄₈₀, for frame size N=480, isherewith provided (960 samples). As explained above, the last values maybe zeros.

+3.538498803770035e−04, +4.595886345493055e−04, +5.595519647227489e−04,+6.718404887323839e−04, +7.919323614002698e−04, +9.254835101516548e−04,+1.069418485858920e−03, +1.227927169310031e−03, +1.398277791978983e−03,+1.584491535026729e−03, +1.783653266717233e−03, +1.999873692180148e−03,+2.230687004705732e−03, +2.479549413444207e−03, +2.744274680521470e−03,+3.028517203082985e−03, +3.329799454594260e−03, +3.651493788541270e−03,+3.991659081158673e−03, +4.353535478916468e−03, +4.734803493964832e−03,+5.139354888882466e−03, +5.564965156664017e−03, +6.015065135312130e−03,+6.487669532207667e−03, +6.986108359341676e−03, +7.507986430774601e−03,+8.056692238103912e−03, +8.629882322202329e−03, +9.230898632007642e−03,+9.857619542786990e−03, +1.051343406844975e−02, +1.119617236058772e−02,+1.190938692951881e−02, +1.265082642578719e−02, +1.342375201935177e−02,+1.422598597626579e−02, +1.506090447446532e−02, +1.592611511144210e−02,+1.682519266177794e−02, +1.775591229287213e−02, 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A numerical example of windowing function, for frame size N=960, isherewith provided 1920 samples). As explained above, the last values maybe zeros.

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+0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00.

While this invention has been described in terms of several embodiments,there are alterations, permutations, and equivalents which fall withinthe scope of this invention. It should also be noted that there are manyalternative ways of implementing the methods and compositions of thepresent invention. It is therefore intended that the following appendedclaims be interpreted as including all such alterations, permutationsand equivalents as fall within the true spirit and scope of the presentinvention.

Bibliography

[1] 3GPP, “Codec for Enhanced Voice Services (EVS); Detailed algorithmicdescription,” [Online]. Available:http://www.3gpp.org/ftp/Specs/archive/26_series/26.445/26445-d00.zip.

[2] P. P. Julien Faure, “Delay-optimized overlap transform,coding/decoding weighting windows”. U.S. Pat. No. 8,847,795, Jun. 282011.

[3] Geiger, Audio Coding based on integer transform, Ilmenau:https://www.db-thueringen.de/receive/dbt_mods_00010054, 2004.

1. An apparatus for encoding an information signal comprising aplurality of frames, the apparatus comprising: a modulated lappedtransform tool for transforming a time domain, TD, representation of theinformation signal, or a processed version thereof, into a frequencydomain, FD, representation, the modulated lapped transform tool beingconfigured to perform a modulated lapped transform analysis using ananalysis windowing function comprising a meandering portion crossing alinear function in correspondence of at least four points; and abitstream writer configured to prepare a bitstream based on the FDrepresentation of the information signal or a processed version thereof,wherein the analysis windowing function is defined so as to beasymmetric, wherein the analysis windowing function is defined so as tobe, in the meandering portion: greater than the linear function in afirst interval between a first crossing point and a second crossingpoint; lower than the linear function in a second interval between thesecond crossing point and a third crossing point; greater than thelinear function in a third interval between the third crossing point anda fourth crossing point, wherein the analysis windowing function isdefined so that the absolute maximum value is in one of the first andthird interval, wherein the linear function is a constant function withconstant value
 1. 2. The apparatus of claim 1, wherein the analysiswindowing function is defined so that the maximum of the analysiswindowing function is less than 25% greater than the value of the linearfunction at the same time instant.
 3. The apparatus of claim 1, whereinthe modulated lapped transform tool is configured to: scale time inputbuffers and/or cosine or sine values with values of the analysiswindowing function.
 4. The apparatus of claim 3, wherein the modulatedlapped transform tool is configured to use input buffers in the form oft(n)=x(Z−N _(F) +n) for n=0 . . . 2N _(F)−1−Z, andt(2N−Z+n)=0 for n=0 . . . Z−1 wherein x(n) is a TD sample of theinformation signal or a processed version of the information signal atthe time instant n, N_(F) is the number of samples processed in oneframe, and Z is the number of leading zeros in modulated lappedtransform window.
 5. The apparatus of claim 1, wherein the modulatedlapped transform tool is configured to perform:${X(k)} = {\sqrt{\frac{2}{N_{F}}}{\sum\limits_{n = 0}^{{2N_{F}} - 1}{{{w_{N}(n)} \cdot {t(n)}}{\cos \left\lbrack {\frac{\pi}{N_{F}}\left( {n + \frac{1}{2} + \frac{N_{F}}{2}} \right)\left( {k + \frac{1}{2}} \right)} \right\rbrack}}}}$for  k = 0  …  N_(F) − 1 where X(k) is the modulated lappedtransform frequency value at a frequency index k, n is the time instant,w_(N) (n) is the analysis windowing function, t(n) is a time inputbuffer, and N_(F) is the number of samples processed in one frame. 6.The apparatus of claim 1, wherein: the analysis windowing functionand/or the synthesis windowing function is defined so that, in themeandering portion, a relative maximum value is in the first or thirdinterval and a relative minimum value is in the second interval.
 7. Theapparatus of claim 1, wherein: the analysis windowing function and/orthe synthesis windowing function is defined so as to present, in themeandering portion, a value greater than the linear function in aninterval comprised of the 30% and 50% of two frames.
 8. The apparatus ofclaim 1, wherein: the analysis windowing function and/or the synthesiswindowing function is defined so that the maximum of the analysiswindowing function and/or the synthesis windowing function is less than25% greater than the value of the linear function at the same timeinstant.
 9. The apparatus of claim 1, wherein: the analysis windowingfunction and/or the synthesis windowing function is defined so that themaximum of the analysis windowing function and/or the synthesiswindowing function is less than 5% greater than the value of the linearfunction at the same time instant.
 10. The apparatus of claim 1,wherein: the analysis windowing function and/or the synthesis windowingfunction is defined so as to present a second numerical differentiationbetween −3*10⁻⁴ and +3*10⁻⁴.
 11. The apparatus of claim 1, wherein: theanalysis windowing function and/or the synthesis windowing function isdefined so as to present a third numerical differentiation between−2*10⁻⁵ and +2*10⁻⁵.
 12. The apparatus of claim 1, wherein: the analysiswindowing function and the synthesis windowing function is defined aretime reversed versions of each other.
 13. The apparatus of claim 1,further comprising: a storage space to store the values of the analysiswindowing function and/or the synthesis windowing function.
 14. Theapparatus of claim 1, wherein the modified lapped transform is amodified discrete cosine transform, MDCT, or a modified discrete sinetransform, MDST, and the inverse modified lapped transform is an inversemodified discrete cosine transform, IMDCT, or inverse modified discretesine transform, IMDST.
 15. The apparatus of claim 1, wherein theanalysis windowing function and/or the synthesis windowing functioncomprises at least a succession, in forward or backward order, formed bythe following values or at least a sub-succession with 10 of thefollowing values if different from 0, with ±1% of tolerance:−7.078546706512391e−04f, −2.098197727900724e−03f,−4.525198076002370e−03f, −8.233976327300612e−03f,−1.337713096257934e−02f, −1.999721557401502e−02f,−2.800909464274782e−02f, −3.721502082245055e−02f,−4.731768261606175e−02f, −5.794654834034055e−02f,−6.867606753531441e−02f, −7.904647440788692e−02f,−8.859705468085925e−02f, −9.688303623049199e−02f,−1.034961241263523e−01f, −1.080766457616878e−01f,−1.103242262600913e−01f, −1.099809851424550e−01f,−1.068172142230882e−01f, −1.006190418791648e−01f,−9.116452506492527e−02f, −7.820617483254730e−02f,−6.146688124166948e−02f, −4.063362855701623e−02f,−1.536329520788766e−02f, +1.470155068746303e−02f,+4.989736509080558e−02f, +9.050369257152079e−02f,+1.366911019414417e−01f, +1.884686389218322e−01f,+2.456456803467095e−01f, +3.077789078889820e−01f,+3.741642373060188e−01f, +4.438114799213576e−01f,+5.154735456539700e−01f, +5.876661722564289e−01f,+6.587619767809000e−01f, +7.270576699841359e−01f,+7.908752989295335e−01f, +8.486643364959733e−01f,+8.991320235484349e−01f, +9.413348145272842e−01f,+9.747634827941575e−01f, +9.994114730415857e−01f,+1.015760373791603e+00f, +1.024736164069697e+00f,+1.027634294456205e+00f, +1.025991493983836e+00f,+1.021427210603284e+00f, +1.015439859549357e+00f,+1.009366925499550e+00f, +1.003508162416449e+00f,+9.988898206257559e−01f, +9.953133902427869e−01f,+9.925943919208190e−01f, +9.905771957917731e−01f,+9.891371616557014e−01f, +9.881790747212391e−01f,+9.876249269174586e−01f, +9.874056275509585e−01f,+9.874524849192456e−01f, +9.876951134084213e−01f,+9.880640617030884e−01f, +9.884926873551375e−01f,+9.889230031022089e−01f, +9.893074965384659e−01f,+9.896146331889107e−01f, +9.898319269347060e−01f,+9.899693102025342e−01f, +9.900603352632121e−01f,+9.901575015155720e−01f, +9.903255289051605e−01f,+9.906303787150326e−01f, +9.911298894709990e−01f,+9.918665491182922e−01f, +9.928619727154252e−01f,+9.941156069136238e−01f, +9.956033775539884e−01f,+9.972793109558521e−01f, +9.990784840729244e−01f,+1.000922365901945e+00f, +1.002728111386909e+00f,+1.004416038098237e+00f, +1.005919224127911e+00f,+1.007189345025525e+00f, +1.008200146369426e+00f,+1.008949493525753e+00f, +1.009458241425143e+00f,+1.009768980817384e+00f, +1.009940336228694e+00f,+1.010039453539107e+00f, +1.010132323996401e+00f,+1.010272524848519e+00f, +1.010494354532353e+00f,+1.010808068774316e+00f, +1.011201071127927e+00f,+1.011641272406023e+00f, +1.012080125934687e+00f,+1.012458183122033e+00f, +1.012706955800289e+00f,+1.012755013843985e+00f, +1.012530134411619e+00f,+1.011962331100864e+00f, +1.010982135506986e+00f,+1.009512438049510e+00f, +1.007460860286395e+00f,+1.004708677491086e+00f, +1.001111413242302e+00f,+9.965041017623596e−01f, +9.907199995730845e−01f,+9.823765865983288e−01f, +9.708821747608998e−01f,+9.546732976073705e−01f, +9.321553861564006e−01f,+9.018003682081348e−01f, +8.623984077953557e−01f,+8.132817365236141e−01f, +7.544551974836834e−01f,+6.866580716267418e−01f, +6.113488038789190e−01f,+5.306181649316597e−01f, +4.471309850999502e−01f,+3.639114681156236e−01f, +2.841647033392408e−01f,+2.110209448747969e−01f, +1.472287968327703e−01f,+9.482665349502291e−02f, +5.482436608328477e−02f,+2.701461405056264e−02f, +9.996743588367519e−03f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f
 16. The apparatus ofclaim 1, wherein the analysis windowing function and/or the synthesiswindowing function comprises at least a succession, in forward orbackward order, formed by the following values or at least asub-succession with 10 of the following values if different from 0, with±1% of tolerance: −4.619898752628163e−04f, −9.747166718929050e−04f,−1.664473096973725e−03f, −2.597106916737789e−03f,−3.806285163352241e−03f, −5.324608721716763e−03f,−7.175885277771099e−03f, −9.382480860899108e−03f,−1.195270300743193e−02f, −1.489528159506296e−02f,−1.820666399965468e−02f, −2.187570925786862e−02f,−2.588471937157619e−02f, −3.020862738245264e−02f,−3.481597793538342e−02f, −3.967067992672979e−02f,−4.472698045914417e−02f, −4.994225863256500e−02f,−5.526334794593565e−02f, −6.063717235243996e−02f,−6.600961519440657e−02f, −7.131966266443390e−02f,−7.651178225890490e−02f, −8.152964005319532e−02f,−8.631137544905677e−02f, −9.080411291245728e−02f,−9.495377758870335e−02f, 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 17. The apparatus ofclaim 1, wherein the analysis windowing function and/or the synthesiswindowing function comprises at least a succession, in forward orbackward order, formed by the following values or at least asub-succession with 10 of the following values if different from 0, with±1% of tolerance: −3.613496418928369e−04f, −7.078546706512391e−04f,−1.074443637110903e−03f, −1.533478537964509e−03f,−2.098197727900724e−03f, −2.778420871815740e−03f,−3.584129920673041e−03f, −4.525198076002370e−03f,−5.609327243712055e−03f, −6.843234536105624e−03f,−8.233976327300612e−03f, −9.785314755557023e−03f,−1.149880303071551e−02f, −1.337713096257934e−02f,−1.542181679511618e−02f, −1.762979910961727e−02f,−1.999721557401502e−02f, −2.252080561390149e−02f,−2.519406300389030e−02f, −2.800909464274782e−02f,−3.095765092956728e−02f, −3.402996266948349e−02f,−3.721502082245055e−02f, −4.050053247568393e−02f,−4.387219218706189e−02f, −4.731768261606175e−02f,−5.082325342672667e−02f, 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 18. The apparatus ofclaim 1, wherein the analysis windowing function and/or the synthesiswindowing function comprises at least a succession, in forward orbackward order, formed by the following values or at least asub-succession with 10 of the following values if different from 0, with±1% of tolerance: −3.021153494057143e−04f, −5.867737487939294e−04f,−8.366504004139796e−04f, −1.126635355725494e−03f,−1.470492941694331e−03f, −1.873473391018495e−03f,−2.339292362082021e−03f, −2.872008069419264e−03f,−3.476256385086407e−03f, −4.155963816705528e−03f,−4.914563787665504e−03f, −5.755172503953251e−03f,−6.680623380533122e−03f, −7.693816924650567e−03f,−8.796760749750191e−03f, −9.990503073705982e−03f,−1.127574117138621e−02f, −1.265334152129685e−02f,−1.412438986522702e−02f, −1.568889620430290e−02f,−1.734512089366117e−02f, −1.909097368362797e−02f,−2.092546711168754e−02f, −2.284684792818856e−02f,−2.485207716234951e−02f, −2.693746704328349e−02f,−2.909952486193999e−02f, 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 19. The apparatus ofclaim 1, wherein the analysis windowing function and/or the synthesiswindowing function comprises at least a succession, in forward orbackward order, formed by the following values or at least asub-succession with 10 of the following values if different from 0, with±1% of tolerance: −2.353032150516754e−04f, −4.619898752628163e−04f,−6.262931535610879e−04f, −7.929180432976445e−04f,−9.747166718929050e−04f, −1.180256894474562e−03f,−1.409209039594871e−03f, −1.664473096973725e−03f,−1.946591608170231e−03f, −2.257081732588478e−03f,−2.597106916737789e−03f, −2.967607624839524e−03f,−3.370454877988472e−03f, −3.806285163352241e−03f,−4.276873767639064e−03f, −4.782469904501813e−03f,−5.324608721716763e−03f, −5.903403814095400e−03f,−6.520419726599805e−03f, −7.175885277771099e−03f,−7.871422820642307e−03f, −8.606586039759667e−03f,−9.382480860899108e−03f, −1.019827182163307e−02f,−1.105520547739066e−02f, −1.195270300743193e−02f,−1.289205910303846e−02f, 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+2.053139897833021e−01f,+1.941021906320988e−01f, +1.831680872008943e−01f,+1.725221947208913e−01f, +1.621735416384834e−01f,+1.521320683467849e−01f, +1.424052801149985e−01f,+1.330015240938615e−01f, +1.239260664828526e−01f,+1.151858295527293e−01f, +1.067840430193724e−01f,+9.872637505002878e−02f, +9.101379000888035e−02f,+8.365057236623055e−02f, +7.663508305536153e−02f,+6.997033405748826e−02f, +6.365188111381365e−02f,+5.768176015814392e−02f, +5.205244216987966e−02f,+4.676538412257621e−02f, +4.180950541438362e−02f,+3.718640251368464e−02f, +3.288072750732215e−02f,+2.889548499582958e−02f, +2.520980565928884e−02f,+2.183057564646272e−02f, +1.872896194002638e−02f,+1.592127815153420e−02f, +1.336381425803020e−02f,+1.108558877807282e−02f, +8.943474189364638e−03f,+6.758124889697787e−03f, +3.504438130619497e−03f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, +0.000000000000000e+00f,+0.000000000000000e+00f, 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 20. The apparatus ofclaim 1, wherein the analysis windowing function and/or the synthesiswindowing function comprises at least a succession, in forward orbackward order, formed by the following values or at least asub-succession with 10 of the following values if different from 0, with±1% of tolerance: −1.596869453315999e−04, −3.021153494057143e−04,−4.142323860121641e−04, −5.058484031439142e−04, −5.867737487939294e−04,−6.666034929771656e−04, −7.499813122143762e−04, −8.366504004139794e−04,−9.273096237893370e−04, −1.023773491532728e−03, −1.126635355725494e−03,−1.235247794937702e−03, −1.349627853510606e−03, −1.470492941694331e−03,−1.598145399394582e−03, −1.732450700747454e−03, −1.873473391018495e−03,−2.021493133113876e−03, −2.176721793926887e−03, −2.339292362082021e−03,−2.509298051958086e−03, −2.686801873698675e−03, −2.872008069419264e−03,−3.065254982136409e−03, −3.266686736002951e−03, −3.476256385086407e−03,−3.694123506258334e−03, −3.920651988943251e−03, −4.155963816705528e−03,−4.399940204176431e−03, −4.652692827862344e−03, −4.914563787665504e−03,−5.185653529531935e−03, −5.465821495567872e−03, −5.755172503953251e−03,−6.054056765330685e−03, −6.362583863449497e−03, −6.680623380533122e−03,−7.008304674718039e−03, −7.346013465075499e−03, −7.693816924650567e−03,−8.051465262788033e−03, −8.419001443263380e−03, −8.796760749750191e−03,−9.184751507972289e−03, −9.582657230262597e−03, −9.990503073705982e−03,−1.040865597419619e−02, −1.083716113754551e−02, −1.127574117138621e−02,−1.172444049589271e−02, −1.218362542421523e−02, −1.265334152129685e−02,−1.313331245904876e−02, −1.362355747138776e−02, −1.412438986522702e−02,−1.463576680723352e−02, −1.515729094197161e−02, −1.568889620430290e−02,−1.623084814367250e−02, −1.678306524600559e−02, −1.734512089366117e−02,−1.791696516251699e−02, −1.849892453107246e−02, −1.909097368362797e−02,−1.969273308274951e−02, −2.030414376031160e−02, −2.092546711168754e−02,−2.155660557689242e−02, −2.219710359529686e−02, 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 21. The apparatus of claim 1, wherein theanalysis windowing function and/or the synthesis windowing functioncomprises at least a succession, in forward or backward order, formed bythe following values or at least a sub-succession with 10 of thefollowing values if different from 0, with ±1% of tolerance:+9.959086585790517e−04, +3.819056787237678e−03, +9.540832613229890e−03,+1.921659800166160e−02, +3.382719081038548e−02, +5.424831667522354e−02,+8.120777668775610e−02, +1.152171887125930e−01, +1.564942331034909e−01,+2.049363422022628e−01, +2.601166575816199e−01, +3.212814164616093e−01,+3.873472997948746e−01, +4.569497078592333e−01, +5.285192958868393e−01,+6.003522489375573e−01, +6.706896380227332e−01, +7.378044458510402e−01,+8.000925313431716e−01, +8.561409184410547e−01, +9.048272294524792e−01,+9.453685031730190e−01, +9.773507430600533e−01, +1.000800872826561e+00,+1.016171590112097e+00, +1.024315247630982e+00, +1.026415431432931e+00,+1.023858366571912e+00, +1.018135705524407e+00, +1.010794822557756e+00,+1.003406509762925e+00, +9.967831265986109e−01, +9.920995520917141e−01,+9.892206942816891e−01, +9.879658322200813e−01, +9.881273531631907e−01,+9.894805541465801e−01, +9.917849916000535e−01, +9.947847580943504e−01,+9.982119669301160e−01, +1.001791235858836e+00, +1.005242583245485e+00,+1.008283053756130e+00, +1.010631281038659e+00, +1.012015300253356e+00,+1.012180753005270e+00, +1.010896765282633e+00, +1.007963362035220e+00,+1.003227255072391e+00, +9.966050551498514e−01, +9.868284225039941e−01,+9.731250287581631e−01, +9.540636479502398e−01, +9.283864275822276e−01,+8.950916858157935e−01, +8.534769362643825e−01, +8.032090930429980e−01,+7.444735201251689e−01, +6.780787033699449e−01, +6.053970453856138e−01,+5.282077505750667e−01, +4.486552956056635e−01, +3.691875990296312e−01,+2.924566408966777e−01, +2.210718537110463e−01, +1.573148583944309e−01,+1.030525757797768e−01, +5.982732244758054e−02, +2.871831923385133e−02,+9.683884928956490e−03, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00.
 22. The apparatus ofclaim 1, wherein the analysis windowing function and/or the synthesiswindowing function comprises at least a succession, in forward orbackward order, formed by the following values or at least asub-succession with 10 of the following values if different from 0, with±1% of tolerance: +6.143388180964179e−04, +1.489582832987000e−03,+2.884104959764029e−03, +4.934298832466617e−03, +7.779130464154915e−03,+1.154910606525086e−02, +1.637155619860352e−02, +2.237116158648752e−02,+2.966159685753317e−02, +3.835663329277230e−02, +4.855610986150206e−02,+6.035055738891727e−02, +7.382288203064732e−02, +8.903563687211119e−02,+1.060356225286319e−01, +1.248534855777947e−01, +1.454931890869180e−01,+1.679435556337752e−01, +1.921728622634411e−01, +2.181238261985594e−01,+2.457259744642953e−01, +2.748839432649996e−01, +3.054824712370942e−01,+3.373873799614014e−01, +3.704415932452488e−01, +4.044749630814483e−01,+4.393004362003260e−01, +4.747225454237193e−01, +5.105341492548225e−01,+5.465201916422433e−01, +5.824658100332457e−01, +6.181452662624718e−01,+6.533411462740817e−01, +6.878367295965062e−01, +7.214176027060971e−01,+7.538887973483771e−01, +7.850546571907628e−01, +8.147397447696774e−01,+8.427819363777799e−01, +8.690376742017057e−01, +8.933935477349644e−01,+9.157483563218768e−01, +9.360270196617569e−01, +9.541731142261065e−01,+9.701635474343885e−01, +9.840036439809510e−01, +9.957199420334376e−01,+1.005374268639838e+00, +1.013046655758663e+00, +1.018843380560658e+00,+1.022896948293643e+00, +1.025355286710874e+00, +1.026382881625701e+00,+1.026155530733488e+00, +1.024853974580724e+00, +1.022664602721801e+00,+1.019779396547454e+00, +1.016391686789653e+00, +1.012697033320358e+00,+1.008885191761748e+00, +1.005378742804807e+00, +1.001563778373068e+00,+9.982531564931281e−01, +9.954346644968789e−01, +9.930950268060122e−01,+9.912170911359961e−01, +9.897805192546195e−01, +9.887624937408933e−01,+9.881383235740961e−01, +9.878819413827574e−01, +9.879662130250981e−01,+9.883630508181326e−01, +9.890434070785485e−01, +9.899772316163624e−01,+9.911334564321237e−01, +9.924800441092685e−01, +9.939841207305906e−01,+9.956121471675398e−01, +9.973300590248015e−01, +9.991033633647473e−01,+1.000897441314013e+00, +1.002677088643863e+00, +1.004407190937699e+00,+1.006052289109999e+00, +1.007576934100958e+00, +1.008945862447015e+00,+1.010124241309341e+00, +1.011077969726137e+00, +1.011773962181442e+00,+1.012180362866919e+00, +1.012266707295288e+00, +1.012004064757857e+00,+1.011365223023975e+00, +1.010324996851905e+00, +1.008860731864438e+00,+1.006952983357691e+00, +1.004586273379809e+00, +1.001749900308864e+00,+9.984386632116344e−01, +9.946500332901397e−01, +9.895756853352172e−01,+9.838303127859196e−01, +9.769999155793757e−01, +9.689141159310996e−01,+9.594038121639412e−01, +9.483086322505029e−01, +9.354860218216989e−01,+9.208101305030523e−01, +9.041732260327581e−01, +8.854882249661838e−01,+8.646864947605046e−01, +8.417237467711145e−01, +8.165875713256009e−01,+7.892986353718001e−01, +7.599171886893816e−01, +7.285474515411827e−01,+6.953282935906302e−01, +6.604334017809461e−01, +6.240661431421666e−01,+5.864461424698465e−01, +5.478160663871147e−01, +5.084499758302218e−01,+4.686361426418982e−01, +4.286789889246253e−01, +3.889032719013045e−01,+3.496431418636314e−01, +3.112360816586544e−01, +2.740128472224535e−01,+2.382847225401666e−01, +2.043379825955252e−01, +1.724305860483632e−01,+1.427939789949265e−01, +1.156385879569741e−01, +9.115821766571995e−02,+6.952749039054593e−02, +5.088975408628225e−02, +3.533430192568954e−02,+2.286680405144430e−02, +1.338005016725895e−02, +6.640506529168652e−03,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00
 23. The apparatus ofclaim 1, wherein the analysis windowing function and/or the synthesiswindowing function comprises at least a succession, in forward orbackward order, formed by the following values or at least asub-succession with 10 of the following values if different from 0, with±1% of tolerance: +5.087227626168386e−04, +9.959086585790517e−04,+1.682208006328800e−03, +2.609697259047744e−03, +3.819056787237678e−03,+5.349319592933909e−03, +7.243906383895192e−03, +9.540832613229890e−03,+1.227637642543709e−02, +1.548950238899404e−02, +1.921659800166160e−02,+2.349369619441617e−02, +2.835199581667961e−02, +3.382719081038548e−02,+3.994939538719628e−02, +4.674775238543380e−02, +5.424831667522354e−02,+6.247770776443612e−02, +7.145835917501348e−02, +8.120777668775610e−02,+9.174400412319896e−02, +1.030764959637497e−01, +1.152171887125930e−01,+1.281665713944242e−01, +1.419264381068653e−01, +1.564942331034909e−01,+1.718593189799504e−01, +1.880134254543744e−01, +2.049363422022628e−01,+2.226123055761096e−01, +2.410151242797736e−01, +2.601166575816199e−01,+2.798871008989962e−01, +3.002880135563586e−01, +3.212814164616093e−01,+3.428208463088390e−01, +3.648596557863134e−01, +3.873472997948746e−01,+4.102294951869188e−01, +4.334494534591082e−01, +4.569497078592333e−01,+4.806696403251166e−01, +5.045473815014847e−01, +5.285192958868393e−01,+5.525196099932443e−01, +5.764872452085427e−01, +6.003522489375573e−01,+6.240509872809882e−01, +6.475182586093196e−01, +6.706896380227332e−01,+6.935029068990036e−01, +7.158927516396895e−01, +7.378044458510402e−01,+7.591787241845952e−01, +7.799586608897265e−01, +8.000925313431716e−01,+8.195318652294690e−01, +8.382288957404715e−01, +8.561409184410547e−01,+8.732316951214179e−01, +8.894702022170831e−01, +9.048272294524792e−01,+9.192736375782965e−01, +9.327940405054362e−01, +9.453685031730190e−01,+9.569883933538136e−01, +9.676486424195593e−01, +9.773507430600533e−01,+9.861027831072527e−01, +9.939122412655677e−01, +1.000800872826561e+00,+1.006787811971719e+00, +1.011901269172423e+00, +1.016171590112097e+00,+1.019636414864842e+00, +1.022336613864005e+00, +1.024315247630982e+00,+1.025621299895396e+00, +1.026303439275662e+00, +1.026415431432931e+00,+1.026007933174836e+00, +1.025137435167917e+00, +1.023858366571912e+00,+1.022226936424625e+00, +1.020300550334848e+00, +1.018135705524407e+00,+1.015792146756340e+00, +1.013325966774524e+00, +1.010794822557756e+00,+1.008265131568879e+00, +1.006046874304407e+00, +1.003406509762925e+00,+1.000977398831985e+00, +9.987704535700208e−01, +9.967831265986109e−01,+9.950118905889862e−01, +9.934523971504882e−01, +9.920995520917141e−01,+9.909475998606236e−01, +9.899902426925508e−01, +9.892206942816891e−01,+9.886318043013834e−01, +9.882160904669929e−01, +9.879658322200813e−01,+9.878730767519871e−01, +9.879296932443894e−01, +9.881273531631907e−01,+9.884575535474619e−01, +9.889115869213529e−01, +9.894805541465801e−01,+9.901553455166457e−01, +9.909266562913843e−01, +9.917849916000535e−01,+9.927206838643636e−01, +9.937239208721489e−01, +9.947847580943504e−01,+9.958931493776203e−01, +9.970389567617592e−01, +9.982119669301160e−01,+9.994020338838508e−01, +1.000598323893564e+00, +1.001791235858836e+00,+1.002969837054169e+00, +1.004123786397111e+00, +1.005242583245485e+00,+1.006315717067918e+00, +1.007332693127034e+00, +1.008283053756130e+00,+1.009156423082384e+00, +1.009942535308151e+00, +1.010631281038659e+00,+1.011212744622770e+00, +1.011677230257499e+00, +1.012015300253356e+00,+1.012217779097186e+00, +1.012275790821109e+00, +1.012180753005270e+00,+1.011924425888915e+00, +1.011498917644724e+00, +1.010896765282633e+00,+1.010110965619444e+00, +1.009135094671655e+00, +1.007963362035220e+00,+1.006590756505588e+00, +1.005013115379014e+00, +1.003227255072391e+00,+1.001231060075500e+00, +9.990235555436858e−01, +9.966050551498514e−01,+9.939894706113089e−01, +9.904539200261149e−01, +9.868284225039941e−01,+9.827716736909488e−01, +9.782206672373213e−01, +9.731250287581631e−01,+9.674323528812744e−01, +9.610947043524248e−01, +9.540636479502398e−01,+9.462952991190324e−01, +9.377489107516087e−01, +9.283864275822276e−01,+9.181762606422500e−01, +9.070861558801854e−01, +8.950916858157935e−01,+8.821696237804294e−01, +8.683025287048570e−01, +8.534769362643825e−01,+8.376852006833730e−01, +8.209275259764013e−01, +8.032090930429980e−01,+7.845450482523652e−01, +7.649554851899686e−01, +7.444735201251689e−01,+7.231348066419057e−01, +7.009860555207412e−01, +6.780787033699450e−01,+6.544686506489734e−01, +6.302212149502727e−01, +6.053970453856138e−01,+5.800715766089168e−01, +5.543129276657669e−01, +5.282077505750727e−01,+5.018369724442092e−01, +4.752902962082383e−01, +4.486552956056652e−01,+4.220281118338883e−01, +3.955057965950340e−01, +3.691875990296320e−01,+3.431732847389720e−01, +3.175633015043183e−01, +2.924566408966782e−01,+2.679463783886042e−01, +2.441231331518492e−01, +2.210718537110466e−01,+1.988719153219592e−01, +1.775967625327044e−01, +1.573148583944310e−01,+1.380903364946733e−01, +1.199837497591550e−01, +1.030525757797769e−01,+8.735085011789188e−02, +7.292811584897502e−02, +5.982732244758056e−02,+4.808178837444506e−02, +3.771135297837851e−02, +2.871831923385135e−02,+2.108352028641225e−02, +1.476289412849005e−02, +9.683884928956495e−03,+5.642168789286858e−03, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00, +0.000000000000000e+00, +0.000000000000000e+00,+0.000000000000000e+00
 24. The apparatus of claim 1, wherein theanalysis windowing function and/or the synthesis windowing functioncomprises at least a succession, in forward or backward order, formed bythe following values or at least a sub-succession with 10 of thefollowing values if different from 0, with ±1% of tolerance:+4.595886345493055e−04, +7.919323614002698e−04, +1.227927169310031e−03,+1.783653266717233e−03, +2.479549413444207e−03, +3.329799454594261e−03,+4.353535478916468e−03, +5.564965156664018e−03, +6.986108359341676e−03,+8.629882322202329e−03, +1.051343406844975e−02, +1.265082642578719e−02,+1.506090447446532e−02, +1.775591229287213e−02, +2.075475983187825e−02,+2.406813715401559e−02, +2.771207863541604e−02, +3.169933248543932e−02,+3.604609640533871e−02, +4.076128638095439e−02, +4.586038120884381e−02,+5.135136676471998e−02, +5.724780220726930e−02, +6.355854744461048e−02,+7.029450733434550e−02, +7.745987198268531e−02, +8.506635369887924e−02,+9.311641620512773e−02, +1.016162955027316e−01, +1.105690806271684e−01,+1.199789286645804e−01, +1.298417294090302e−01, +1.401623800497866e−01,+1.509371564593891e−01, +1.621632295622287e−01, +1.738354123649302e−01,+1.859520359191026e−01, +1.985008828937603e−01, +2.114778554475382e−01,+2.248732557074316e−01, +2.386763947872762e−01, +2.528729453658238e−01,+2.674547009618951e−01, +2.824031465430401e−01, +2.977050145264297e−01,+3.133419120661713e−01, +3.292976696294886e−01, +3.455490160824131e−01,+3.620795045342974e−01, +3.788648665671841e−01, +3.958851576591690e−01,+4.131143794748322e−01, +4.305308301005456e−01, +4.481076715576617e−01,+4.658227790464821e−01, +4.836466393241829e−01, +5.015564851667653e−01,+5.195228071176610e−01, +5.375197039843709e−01, +5.555183841040963e−01,+5.734957812557457e−01, +5.914186654649489e−01, +6.092622887527459e−01,+6.269981160888640e−01, +6.446002007776794e−01, +6.620384583071039e−01,+6.792906550106088e−01, +6.963256426589250e−01, +7.131194393772130e−01,+7.296469905863920e−01, +7.458864594794676e−01, 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 25. The apparatus ofclaim 1, wherein the analysis windowing function and/or the synthesiswindowing function comprises at least a succession, in forward orbackward order, formed by the following values or at least asub-succession with 10 of the following values if different from 0, with±1% of tolerance: +4.090106504820579e−04, +6.143388180964179e−04,+8.571759876954877e−04, +1.147015057857495e−03, +1.489582832987000e−03,+1.889770382231583e−03, +2.353000800169909e−03, +2.884104959764029e−03,+3.488213786635855e−03, +4.170040431489613e−03, +4.934298832466617e−03,+5.787076505403503e−03, +6.733811743137561e−03, +7.779130464154915e−03,+8.927044958757816e−03, +1.018202888968871e−02, +1.154910606525086e−02,+1.303349217699797e−02, +1.463951288465963e−02, +1.637155619860352e−02,+1.823455383898077e−02, +2.023309488998589e−02, +2.237116158648752e−02,+2.465237348403478e−02, +2.708101935270475e−02, +2.966159685753317e−02,+3.239884850877327e−02, +3.529601774976465e−02, 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 26. The apparatus of claim 1, wherein theanalysis windowing function and/or the synthesis windowing functioncomprises at least a succession, in forward or backward order, formed bythe following values or at least a sub-succession with 10 of thefollowing values if different from 0, with ±1% of tolerance:+3.821992968116373e−04, +5.337292962876158e−04, +7.010318350640769e−04,+8.910357994632938e−04, +1.107726421735994e−03, +1.354305556720234e−03,+1.632968062524617e−03, +1.944392065549996e−03, +2.291251920978034e−03,+2.676357471188558e−03, +3.102160324883627e−03, +3.569302917352426e−03,+4.080147004222224e−03, +4.637409387087253e−03, +5.243697651112320e−03,+5.900394401532485e−03, +6.609908124535202e−03, +7.375089966000295e−03,+8.197610411215884e−03, +9.078195201482999e−03, +1.001888876231112e−02,+1.102272421949215e−02, +1.209201783148431e−02, +1.322773320467288e−02,+1.443170227861997e−02, +1.570673344965518e−02, +1.705481769961258e−02,+1.847711280171765e−02, +1.997546609433320e−02, 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 27. The apparatus of claim 1, wherein theanalysis windowing function and/or the synthesis windowing functioncomprises at least a succession, in forward or backward order, formed bythe following values or at least a sub-succession with 10 of thefollowing values if different from 0, with ±1% of tolerance:+3.538498803770035e−04, +4.595886345493055e−04, +5.595519647227489e−04,+6.718404887323839e−04, +7.919323614002698e−04, +9.254835101516548e−04,+1.069418485858920e−03, +1.227927169310031e−03, +1.398277791978983e−03,+1.584491535026729e−03, +1.783653266717233e−03, +1.999873692180148e−03,+2.230687004705732e−03, +2.479549413444207e−03, +2.744274680521470e−03,+3.028517203082985e−03, +3.329799454594260e−03, +3.651493788541270e−03,+3.991659081158673e−03, +4.353535478916468e−03, +4.734803493964832e−03,+5.139354888882466e−03, +5.564965156664017e−03, +6.015065135312130e−03,+6.487669532207667e−03, +6.986108359341676e−03, +7.507986430774601e−03,+8.056692238103912e−03, +8.629882322202329e−03, 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The apparatus of claim 1, wherein the analysis windowing functionand/or the synthesis windowing function comprises at least a succession,in forward or backward order, formed by the following values or at leasta sub-succession with 10 of the following values if different from 0,with ±1% of tolerance: +3.235700349233950e−04, +3.821992968116373e−04,+4.346763076430269e−04, +4.841399974556557e−04, +5.337292962876158e−04,+5.864735900759105e−04, +6.428827504324996e−04, +7.010318350640768e−04,+7.608344672311270e−04, +8.240821499679188e−04, +8.910357994632938e−04,+9.604008359868557e−04, +1.032187650884365e−03, +1.107726421735994e−03,+1.187132578842487e−03, +1.269300730654068e−03, +1.354305556720234e−03,+1.443403284874795e−03, +1.536663296357133e−03, +1.632968062524617e−03,+1.732380147404573e−03, +1.836145247521724e−03, +1.944392065549996e−03,+2.056125836181849e−03, +2.171386021601719e−03, +2.291251920978034e−03,+2.415808383814314e−03, +2.544144440874477e−03, 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 29. An apparatus according to claim 1, whereinthe analysis windowing function and/or synthesis windowing function isdefined so as to present a first numerical differentiation between −0.01and +0.01.
 30. An apparatus according to claim 1, wherein the analysiswindowing function and/or synthesis windowing function is defined so asto present a second numerical differentiation between −10⁻⁴ and +10⁻⁴.31. An apparatus according to claim 1, wherein the analysis windowingfunction and/or synthesis windowing function is defined so as to presenta second numerical differentiation between −10⁻⁵ and +10⁻⁵.
 32. Anapparatus according to claim 1, wherein the information signal is anaudio signal.
 33. A method for encoding an information signal comprisinga plurality of frames, the method comprising: performing a modifiedcosine transformation, MDCT, analysis for transforming a time domain,TD, representation of the information signal, or a processed versionthereof, into a frequency domain, FD, representation, using an analysiswindowing function comprising a meandering portion which passes a linearfunction in correspondence of at least four points, wherein the analysiswindowing function is defined so as to be asymmetric, wherein theanalysis windowing function is defined so as to be, in the meanderingportion: greater than the linear function in a first interval between afirst crossing point and a second crossing point; lower than the linearfunction in a second interval between the second crossing point and athird crossing point; greater than the linear function in a thirdinterval between the third crossing point and a fourth crossing point,wherein the analysis windowing function is defined so that the absolutemaximum value is in one of the first and third interval, wherein thelinear function is a constant function with constant value
 1. 34. Anon-transitory digital storage medium having a computer program storedthereon to perform the method for encoding an information signalcomprising a plurality of frames, said method comprising: performing amodified cosine transformation, MDCT, analysis for transforming a timedomain, TD, representation of the information signal, or a processedversion thereof, into a frequency domain, FD, representation, using ananalysis windowing function comprising a meandering portion which passesa linear function in correspondence of at least four points, wherein theanalysis windowing function is defined so as to be asymmetric, whereinthe analysis windowing function is defined so as to be, in themeandering portion: greater than the linear function in a first intervalbetween a first crossing point and a second crossing point; lower thanthe linear function in a second interval between the second crossingpoint and a third crossing point; greater than the linear function in athird interval between the third crossing point and a fourth crossingpoint, wherein the analysis windowing function is defined so that theabsolute maximum value is in one of the first and third interval,wherein the linear function is a constant function with constant value1, when said computer program is run by a computer.